# Nondefectivity of invariant secant varieties

Alexander Taveira Blomenhofer  Centrum Wiskunde en Informatica (CWI), Science
Park 123, 1098XG Amsterdam, The Netherlands<EMAIL_ADDRESS>and  Alex Casarotti
Università di Bologna, Piazza di Porta S. Donato 5, (40126) Bologna, Italy
<EMAIL_ADDRESS>
###### Abstract.

We show that a large class of secant varieties is nondefective. In particular,
we positively resolve most cases of the Baur-Draisma-de Graaf conjecture on
Grassmannian secants in large dimensions. Our result improves the known bounds
on nondefectivity for various other secant varieties, including Chow
varieties, Segre-Veronese varieties and Gaussian moment varieties. We also
give bounds for identifiability and the generic ranks.

###### 2020 Mathematics Subject Classification:

primary: 14N07, 15A72

##  1\. Introduction

Our starting point is the empirical observation that many secant varieties
behave highly “regular” with respect to dimension. If one picks a “reasonable”
base variety $V$ of small dimension $N$ in a space $S^{d}(\mathbb{C}^{n})$ of
polynomials, then very often, the first secant varieties $\sigma_{m}(V)$ will
have the _expected dimension_ for small $m$, which, in affine notation, equals
$mN$.

Conversely, for large $m$, the secants of a nondegenerate variety $V$ will
fill up the entire space $S^{d}(\mathbb{C}^{n})$. The smallest $m$ such that
the $m$-th secant fills the whole space is called the _generic $V$-rank_,
denoted $m^{\circ}$. Frequently, the lower bound

$\displaystyle m^{\circ}\geq\frac{\dim S^{d}(\mathbb{C}^{n})}{\dim V},$ (1)

obtained from counting parameters, is a very close estimate for the true value
of $m^{\circ}$. One of the first proven results of this type was the
Alexander-Hirschowitz theorem [15], which concerned decompositions

$\displaystyle f=\sum_{i=1}^{m}\ell_{i}^{d}$ (2)

of a $d$-form $f$ in $n$ variables as a sum of $d$-th powers of linear forms
$\ell_{1},\ldots,\ell_{m}$. Decompositions like Equation 2 are called
$1$-Waring decompositions. The value $m$ is called the $1$-Waring _rank_ of
the decomposition. In this example, $V=\nu_{d}(\mathbb{C}^{n})$ is the
degree-$d$ Veronese embedding of $\mathbb{C}^{n}$. Any form $f$ with a
decomposition like Equation 2 lies in the $m$-th secant variety
$\sigma_{m}(V)$. The Alexander-Hirschowitz theorem states that for all but
finitely many $(n,d)$ with $n\geq 2,d\geq 3$, the generic $1$-Waring rank is
the rounded-up value of $\frac{\dim S^{d}(\mathbb{C}^{n})}{\dim V}$, and that
most secants of rank smaller than $m^{\circ}$ have the expected dimension.

If the linear forms $\ell_{i}$ are replaced by forms $q_{i}$ of higher degree
$k$, then one obtains the notion of a $k$-Waring decomposition

$\displaystyle f=\sum_{i=1}^{m}q_{i}^{d}$ (3)

of a form $f$ of degree $kd$. Those higher-order Waring decompositions can be
traced back a long time. E.g., Ramanujan [22] posed a famous problem about
sums of cubes of quadratics in 1913. In recent times, the study of Waring
decomposition was fueled by connections to Machine Learning [4] and circuit
complexity [14]. Defectivity and identifiability properties with respect to
$k$-Waring problem have been treated in [8].

Yet another type of decompositions within $S^{d}(\mathbb{C}^{n})$ are Chow
decompositions. Here, one aims to write a $d$-form

$\displaystyle f=\sum_{i=1}^{m}\ell_{i,1}\cdot\ldots\cdot\ell_{i,d}$ (4)

as a sum of $d$-fold products of linear forms $\ell_{i,j}$. All of these
decompositions have one thing in common: They start with an irreducible,
affine cone $V\subseteq S^{d}(\mathbb{C}^{n})$ that is closed under the action
of $\operatorname{GL}(\mathbb{C}^{n})$. Subsequently, they try to write an
element $f$ of $S^{d}(\mathbb{C}^{n})$ as a sum of finitely many elements of
$V$.

Therefore, one may produce more problems of this type, by continuing the list
of $\operatorname{GL}$-invariant subvarieties of $S^{d}(\mathbb{C}^{n})$.
Machine Learning motivates to look at _Gaussian moment varieties_ , since
their secants are important to Gaussian mixture models.

A very similar behaviour is encountered, if instead of the symmetric powers,
one looks at canonical subvarieties of the alternating powers
$\bigwedge^{d}(\mathbb{C}^{n})$. The most natural choice is the Grassmannian
variety $\operatorname{Gr}(n,d)$ of $d$-dimensional subspaces of
$\mathbb{C}^{n}$. It may be embedded into
$\mathbb{P}(\bigwedge^{d}(\mathbb{C}^{n}))$ via the _Plücker map_

$\displaystyle\langle u_{1},\ldots,u_{d}\rangle\mapsto[u_{1}\wedge\ldots\wedge
u_{d}],$ (5)

which sends a basis $u_{1},\ldots,u_{d}$ of the space to the projective
equivalence class of its wedge product. Secant varieties of Grassmannians are
of significance to coding theory, as outlined by Baur, Draisma and de Graaf
[16].

### Contributions

We give a unified treatment and bounds for nondefectivity and the generic rank
in all the cases mentioned above. For Grassmannians, Chow varieties, Gaussian
moment and Segre-Veronese varieties, our new bounds improve significantly on
the best known ones and are asymptotically optimal for large $n$. The common
structure in the examples above is the presence of a group action, e.g.,
$G=\operatorname{GL}(\mathbb{C}^{n})$, and of an irreducible $G$-module
$\mathcal{L}$, e.g., $\mathcal{L}=S^{d}(\mathbb{C}^{n})$ or
$\mathcal{L}=\bigwedge^{d}(\mathbb{C}^{n})$. We leverage invariant-theoretic
techniques to show that all of the above-mentioned varieties, and many more,
will have most of their secant dimensions as expected. Our main result is
Theorem 3.4, but the applications all follow from its Corollary, Theorem 1.1,
which we will state here and prove in 3.5.

######  1.1 Theorem.

Let $G$ be a group and $V$ an $N$-dimensional, irreducible affine cone in an
irreducible $G$-module $\mathcal{L}$. If $V$ is closed under the $G$-action,
then it holds that

  1. (1)

$V$ is $m$-nondefective for all $m\leq\frac{\dim\mathcal{L}}{N}-N$.

  2. (2)

The generic rank (with respect to $V$) is between $\frac{\dim\mathcal{L}}{N}$
and $\frac{\dim\mathcal{L}}{N}+N$.

Despite its generality, Theorem 1.1 gives better bounds than quite a few
specialized results for specific varieties. In Section 4, we list cases where
our new result improves on the currently best-known bounds for nondefectivity,
identifiability and/or the generic rank. In many of those cases, our bounds
are asymptotically optimal for large $n$. In particular, we prove most cases
of the Baur-Draisma-de Graaf conjecture, stated in 4.2.

### Organization

We prove our main result in Section 3. In Section 4, we apply it to a plethora
of different varieties, where we can obtain improved bounds on nondefectivity
and the generic rank.

### Techniques and Acknowledgments

In the 1980’s, Ådlandsvik examined varieties, which have a large number of
defective secant varieties, before they become (projectively) a cone. In 2015,
Nenashev resolved a large number of cases of Fröberg’s conjecture for forms of
equal degree via a very short, clean and elegant argument. Both independently
made observations, which are special cases of our main technical result, Lemma
3.2. In this work, we present a generalization of both.

We are grateful to Edoardo Ballico for pointing us towards the work of Bjørn
Ådlandsvik, and to Vincenzo Galgano, who suggested the application to spinor
varieties. We also wish to thank Alessandro Oneto for inviting the authors to
the University of Trento and Nick Vannieuwenhoven for telling us about the
(non-)weak defectivity of Chow varieties.

##  2\. Preliminaries

Let us briefly recall the main notions used in this paper. Most of our
varieties will be affine cones $V$ in a linear space $\mathcal{L}$, i.e.
$\mathbb{C}\cdot V\subseteq V$. The corresponding projective varieties we
denote by $\mathbb{P}(V)\subseteq\mathbb{P}(\mathcal{L})$. Frequently, we will
also have a group $G$ acting on the space $\mathcal{L}$, giving it the
structure of a $G$-module. We say that a variety $V$ is $G$-_invariant_ , if
for all $v\in V$ and $g\in G$, it holds $gv\in V$. We use two different
notions of irreducibility, that are not to be confused: A variety is called
irreducible, if it cannot be covered by the union of two proper subvarieties.
A $G$-module $\mathcal{L}$ is called irreducible, if $\mathcal{L}$ is not the
zero module and $\mathcal{L}$ does not contain any proper nonzero
$G$-submodules. The set of smooth points of a variety $V$ is denoted
$V_{\operatorname{reg}}$. For every smooth point $x\in V$ we denote by
$T_{x}V$ the tangent space to $V$ at $x$. If $x_{1},\dots,x_{m}$ are smooth
points of $V$ we denote by $\langle
T_{x_{1}}V_{1},\ldots,T_{x_{m}}V_{m}\rangle$ the linear span of the
corresponding tangent spaces. The space of $d$-forms on a vector space
$\mathcal{L}$ is denoted $S^{d}(\mathcal{L})$. A special case is the dual
space $\mathcal{L}^{\vee}=S^{1}(\mathcal{L})$.

######  Definition 1.

The _$m$ -th secant variety_ $\sigma_{m}V$ of the affine cone $V$ is the
Zariski closure of

$\displaystyle\\{x_{1}+\ldots+x_{m}\mid x_{1},\ldots,x_{m}\in V\\}.$ (6)

The $m$-th secant variety has the _expected dimension_
$e_{m}(V):=\min\\{mN,\dim\mathcal{L}\\}$. If the dimension of $\sigma_{m}V$ is
smaller than expected, we say that $V$ is $m$-_defective_. Otherwise, $V$ is
called $m$-_nondefective_. We say that $V$ is $m$-identifiable, if the general
point $p\in\sigma_{m}V$ has, up to permutation, a unique representation
$p=x_{1}+\ldots+x_{m}$, where $x_{1},\ldots,x_{m}\in V$. Secant varieties are
a special case of joins: The _join_ of affine cones $V_{1},\ldots,V_{m}$ is
defined as the closure of $\\{x_{1}+\ldots+x_{m}\mid x_{1}\in
V_{1},\ldots,x_{m}\in V_{m}\\}$. We denote it by $J(V_{1},\ldots,V_{m})$.

The main tool in order to compute the dimension of joins and secant varieties
dates back to Terracini [24] and can be summarized as follows:

######  2.1 Lemma (Terracini, see [24]).

For general $x_{1}\in V_{1},\ldots,x_{m}\in V_{m}$ and general $z\in\langle
x_{1},\ldots,x_{m}\rangle$, the tangent space to the join
$J(V_{1},\ldots,V_{m})$ at $z$ equals $\langle
T_{x_{1}}V_{1},\ldots,T_{x_{m}}V_{m}\rangle$.

Moreover, for smooth points $x_{1}\in V_{1},\ldots,x_{m}\in V_{m}$, and any
$z\in\langle x_{1},\ldots,x_{m}\rangle$, $\langle
T_{x_{1}}V,\ldots,T_{x_{m}}V\rangle$ is still contained in the tangent to
$J(V_{1},\ldots,V_{m})$ at $z$.

######  2.2 Proposition.

The _apex space_ of an (irreducible) affine cone $V$ is defined as

$\displaystyle\operatorname{ap}(V)=\\{x\in V\mid V+x=V\\}.$ (7)

For brevity, we also denote
$\operatorname{ap}_{m}(V):=\operatorname{ap}(\sigma_{m}V)$. It holds that

$\displaystyle\operatorname{ap}_{m}(V)=\bigcap_{(x_{1},\ldots,x_{m})\in\mathcal{U}_{m,V}}\langle
T_{x_{1}}V,\ldots,T_{x_{m}}V\rangle,$ (8)

where $\mathcal{U}_{V,m}$ denotes the set

$\displaystyle\\{x\in V_{\mathrm{reg}}^{m}\mid\langle
T_{x_{1}}V,\ldots,T_{x_{m}}V\rangle=T_{x_{1}+\ldots+x_{m}}\sigma_{m}V\text{
has generic dimension}\\}.$ (9)

###### Proof.

For the inclusion from left to right, let $p\in\operatorname{ap}_{m}(V)$ and
let $(x_{1},\ldots,x_{m})\in\mathcal{U}_{V,m}$. By definition of the $m$-apex,
it holds $J(\sigma_{m}V,\langle p\rangle)=\sigma_{m}V$. Terracini’s Lemma
applied to the join yields $\langle
T_{x_{1}}V,\ldots,T_{x_{m}}V,p\rangle\subseteq\langle
T_{x_{1}}V,\ldots,T_{x_{m}}V\rangle$. Thus, $p$ lies in $\langle
T_{x_{1}}V,\ldots,T_{x_{m}}V\rangle$.

Conversely, let $p$ in the right hand side. Choose a general point
$x\in\mathcal{U}_{V,m}$. Again by Terracini’s Lemma, equality $\langle
T_{x_{1}}V,\ldots,T_{x_{m}}V,p\rangle=\langle
T_{x_{1}}V,\ldots,T_{x_{m}}V\rangle$ holds. Since $x$ is general, thus the
dimensions of $J(\sigma_{m}V,\langle p\rangle)$ and of $\sigma_{m}V$ are
equal. By irreducibility, the varieties are equal. Thus
$p\in\operatorname{ap}_{m}(V)$. ∎

######  2.3 Remark.

The apex space is a linear space satisfying $\operatorname{ap}(V)\subseteq V$.
If $\mathcal{L}$ is a $G$-module and $V$ is $G$-invariant, then
$\operatorname{ap}(V)$ is a $G$-submodule of $\mathcal{L}$.

##  3\. The stationarity lemma

Let $G$ a group acting on the linear space $\mathcal{L}$, such that
$\mathcal{L}$ is an irreducible $G$-module. We consider irreducible algebraic
varieties $V,V^{\prime}$, which are also embedded in some $G$-modules and
which are closed under the $G$-action. Furthermore, we consider
$G$-equivariant (algebraic) maps

$\displaystyle T\colon\mathbb{P}(V)\to\mathrm{Gr}(N,\mathcal{L}),[x]\mapsto
T_{x},$ (10) $\displaystyle
U\colon\mathbb{P}(V^{\prime})\to\mathrm{Gr}(N^{\prime},\mathcal{L}),[y]\mapsto
U_{y},$ (11)

By $\operatorname{Gr}(N,\mathcal{L})$, we denote the Grassmannian variety of
$N$-dimensional subspaces of $\mathcal{L}$. Note that $G$ acts canonically on
the Grassmannian varieties of $\mathcal{L}$.

Let $\mathcal{U}_{T,m}=\\{(x_{1},\ldots,x_{m})\mid\langle
T_{x_{1}},\ldots,T_{x_{m}}\rangle\text{ is of generic dimension}\\}$. In
analogy to 2.2, we define the $m$-th apex of the map $T$ to be

$\displaystyle\operatorname{ap}_{m}(T):=\bigcap_{(x_{1},\ldots,x_{m})\in\mathcal{U}_{T,m}}\langle
T_{x_{1}},\ldots,T_{x_{m}}\rangle$ (12)

Taking $T\colon V_{\operatorname{reg}}\to\operatorname{Gr}(\dim
V,\mathcal{L})$ as the tangent map, one recovers the apex space from 2.2.

######  3.1 Lemma.

Let $\mathcal{U}$ a dense open subset of $\mathcal{U}_{T,m}$. Then

$\displaystyle\operatorname{ap}_{m}(T)=\bigcap_{(x_{1},\ldots,x_{m})\in\mathcal{U}}\langle
T_{x_{1}},\ldots,T_{x_{m}}\rangle$ (13)

###### Proof.

The inclusion from left to right is clear. Now, let $p$ be contained in the
right hand side. Then, $p$ lies in $\langle T_{x_{1}},\ldots,T_{x_{m}}\rangle$
for all $x=(x_{1},\ldots,x_{m})\in\mathcal{U}$. Let $R$ denote the generic
dimension of $\langle T_{x_{1}},\ldots,T_{x_{m}}\rangle$. For all
$x\in\mathcal{U}_{T,m}$, by the Plücker embedding, we can represent $\langle
T_{x_{1}},\ldots,T_{x_{m}}\rangle$ as a projective equivalence class of an
element $t(x_{1},\ldots,x_{m})\in\bigwedge^{R}(\mathbb{C}^{n})$, where $t$
depends polynomially on $(x_{1},\ldots,x_{m})$. Define $f(p,x)=p\wedge
t(x_{1},\ldots,x_{m})$. Clearly, $f(p,x)$ is the zero tensor for all
$x\in\mathcal{U}$. By continuity, it is constantly zero. But for
$x\in\mathcal{U}_{T,m}$, the identity $p\wedge t(x_{1},\ldots,x_{m})=0$
implies that $p\in\langle T_{x_{1}},\ldots,T_{x_{m}}\rangle$. ∎

The next Lemma is a generalization of previous results from [1] and [19]. For
any $m\in\mathbb{N}$, define:

$\displaystyle a_{m}:=\dim(\langle T_{x_{1}},\ldots,T_{x_{m}}\rangle\cap
U_{y})$ (14)

where $x_{1},\ldots,x_{m}\in V,y\in V^{\prime}$ are generic points. Clearly,
we have that

$\displaystyle 0=a_{0}\leq a_{1}\leq a_{2}\leq\ldots$ (15)

######  3.2 Lemma (Stationarity Lemma).

Let $\mathcal{L}$ an irreducible $G$-module and $V$, $V^{\prime}$, $T$, $U$,
$(a_{m})_{m}$ as introduced above. If $m\in\mathbb{N}$ is such that
$a_{m+1}=a_{m}\neq 0$, then $\langle
T_{x_{1}},\ldots,T_{x_{m}}\rangle=\mathcal{L}$ and $a_{m}=N^{\prime}$.

###### Proof.

Consider generic points
$x_{1},\ldots,x_{m},x_{1}^{\prime},\ldots,x_{m}^{\prime}\in V$, and generic
$y\in V^{\prime}$. We know that

$\displaystyle\dim\langle T_{x_{1}},\ldots,T_{x_{m}}\rangle\cap
U_{y}=a_{m}=a_{m+1}=\dim\langle
T_{x_{1}},\ldots,T_{x_{m}},T_{x_{m+1}}\rangle\cap U_{y}.$

Therefore, it also holds

$\displaystyle\dim\langle T_{x_{1}},\ldots,T_{x_{m}}\rangle\cap
U_{y}=\dim\langle T_{x_{1}},\ldots,T_{x_{m}},T_{x_{m}^{\prime}}\rangle\cap
U_{y}.$ (16)

The space on the left-hand side is a subspace of the space on the right-hand
side, of same dimension. Hence, they are equal:

$\displaystyle\langle T_{x_{1}},\ldots,T_{x_{m}}\rangle\cap U_{y}=\langle
T_{x_{1}},\ldots,T_{x_{m}},T_{x_{m}^{\prime}}\rangle\cap U_{y}.$ (17)

We may apply the same argument with exchanged roles of $x_{m}$ and
$x_{m}^{\prime}$ to obtain

$\langle T_{x_{1}},\ldots,T_{x_{m-1}},T_{x_{m}^{\prime}}\rangle\cap
U_{y}=\langle T_{x_{1}},\ldots,T_{x_{m}},T_{x_{m}^{\prime}}\rangle\cap U_{y}.$

Together, this implies

$\displaystyle\langle
T_{x_{1}},\ldots,T_{x_{m-1}},T_{x_{m}^{\prime}}\rangle\cap U_{y}=\langle
T_{x_{1}},\ldots,T_{x_{m}}\rangle\cap U_{y}$ (18)

We thus showed that swapping $x_{m}$ with $x_{m}^{\prime}$ does not change the
space from Equation 18. Repeating the procedure with all pairs of
$x_{i},x_{i}^{\prime}$, we obtain

$\displaystyle\langle T_{x_{1}^{\prime}},\ldots,T_{x_{m}^{\prime}}\rangle\cap
U_{y}=\langle T_{x_{1}},\ldots,T_{x_{m}}\rangle\cap U_{y}$ (19)

In other words, any point $p$ in the set from Equation 19 lies in $\langle
T_{x_{1}},\ldots,T_{x_{m}}\rangle$ for all general choices of
$x_{1},\ldots,x_{m}$. Thus,

$\displaystyle\langle T_{x_{1}},\ldots,T_{x_{m}}\rangle\cap
U_{y}=\operatorname{ap}_{m}(T)\cap U_{y},$ (20)

by Lemma 3.1. In particular, the $m$-th apex of $T$ is not the zero space,
since $a_{m}>0$. The space $\operatorname{ap}_{m}(T)$ is $G$-invariant and
$\mathcal{L}$ is an irreducible $G$-module. Therefore, it follows that
$\mathcal{L}=\operatorname{ap}_{m}(T)\subseteq\langle
T_{x_{1}},\ldots,T_{x_{m}}\rangle$. ∎

######  3.3 Remark.

Similar arguments were present in the work of Ådlandsvik [1] and Nenashev
[19]. Ådlandsvik considered the case where $T=U$ is the tangent map of a
variety $V$, which associates to $x\in V_{\operatorname{reg}}$ the tangent
space $T_{x}V$. See [1, Proposition 2.1(ii)]. He proves a similar stationarity
lemma, but he did not endow $\mathcal{L}$ with a group action. As a result, he
arrives at the weaker conclusion that at a stationarity point $0\neq
a_{m}=a_{m+1}$, the apex $\operatorname{ap}_{m}(T)$ must not be the zero
space.

Independently, Nenashev proved in [19] a result about the Hilbert series of
general ideals. It is also a special case of our result, obtained if one takes
$V$ to be any $\operatorname{GL}$-invariant variety in $S^{d}(\mathbb{C}^{n})$
and $\mathcal{L}=S^{d+k}(\mathbb{C}^{n})$, which is an irreducible
$\operatorname{GL}$-module. Here, the maps $T=U$ are equal and given by
$T_{x}=(x)_{d+k}$. In other words, a form $x\in V$ is mapped to some graded
component of the principal ideal it generates.

######  3.4 Theorem.

Let $\mathcal{L}$ be an irreducible $G$-module, $V,V^{\prime}$ irreducible
$G$-invariant affine cones of dimensions $N$ and $N^{\prime}$, respectively.
Let $T\colon\mathbb{P}(V)\to\operatorname{Gr}(N,\mathcal{L})$ and
$U\colon\mathbb{P}(V^{\prime})\to\operatorname{Gr}(N^{\prime},\mathcal{L})$ be
$G$-equivariant maps. Then, there exists an interval
$\mathcal{I}\subseteq\mathbb{N}$ of length at most $N^{\prime}$, such that for
all $m\notin\mathcal{I}$, and generic $x_{1},\ldots,x_{m}\in V$, generic $y\in
V^{\prime}$, either one of the following holds true:

  1. (1)

$\langle T_{x_{1}},\ldots,T_{x_{m}}\rangle\cap U_{y}=\\{0\\}$ or

  2. (2)

$\langle T_{x_{1}},\ldots,T_{x_{m}}\rangle=\mathcal{L}.$

###### Proof.

There exists a smallest $m_{1}\in\mathbb{N}$ such that $\langle
T_{x_{1}},\ldots,T_{x_{m_{1}}}\rangle=\mathcal{L}$. There also exists a
largest $m_{0}\in\mathbb{N}_{0}$ such that $\langle
T_{x_{1}},\ldots,T_{x_{m_{0}}}\rangle\cap U_{y}=\\{0\\}$. The claim holds true
for all $m\notin\mathcal{I}:=\\{m_{0}+1,\ldots,m_{1}\\}$. The length of this
interval is $m_{1}-m_{0}$. By Lemma 3.2, we know that the dimension of the
intersection is strictly increasing between $m_{0}$ and $m_{1}$, and for
$m_{1}$, the intersection equals $U_{y}$. Therefore,

$\displaystyle m_{1}-m_{0}\leq a_{m_{1}}-a_{m_{0}}=\dim U_{y}=N^{\prime}.$
(21)

This asserts the claim. ∎

In the special case $T=U$, it is possible to deduce explicit bounds for the
quantities $m_{0}$ and $m_{1}$, which occur in the proof of Theorem 3.4. In
the following, we show this for the even more special case, where $T=U$ is the
tangent map of a variety. In this way, we obtain a very general criterion for
secant nondefectivity and estimates for the generic rank with respect to a
large class of base varieties.

######  3.5 Corollary.

Let $V$ be an $N$-dimensional, irreducible subvariety of the irreducible
$G$-module $\mathcal{L}$. Assume $V$ is $G$-invariant. Then,

  1. (1)

$V$ is $m$-nondefective for all $m\leq\frac{\dim\mathcal{L}}{N}-N$.

  2. (2)

The generic rank of $\mathcal{L}$ with respect to $V$ is at most
$m^{\circ}\leq\frac{\dim\mathcal{L}}{N}+N$.

###### Proof.

In Theorem 3.4, take $T=U$ as the tangent map

$\displaystyle T\colon
V_{\operatorname{reg}}\to\operatorname{Gr}(\mathcal{L},N),x\mapsto T_{x}V,$
(22)

which takes a point $x\in V$ to its tangent space. We have $N=N^{\prime}=\dim
V$ in the notation of Theorem 3.4. Let $m_{0}$ and $m_{1}$ be as in Theorem
3.4. From counting parameters, we have the obvious bounds
$N(m_{0}+1)\leq\dim\mathcal{L}$ and $Nm_{1}\geq\dim\mathcal{L}$. Using these
together with Equation 21, we obtain

$\displaystyle m_{0}\geq m_{1}-N\geq\frac{\dim\mathcal{L}}{N}-N,$ (23)

and

$\displaystyle m_{1}\leq
m_{0}+N\leq\frac{\dim\mathcal{L}-N}{N}+N=\frac{\dim\mathcal{L}}{N}+N-1.$ (24)

and we obtain that a sum of $m$ tangent spaces is direct, as long as
$m\leq\frac{\dim\mathcal{L}}{N}-N$. Similarly, we also obtain from Theorem 3.4
that if $m\geq\frac{\dim\mathcal{L}}{N}+N$, then the $m$-th secant variety
$\sigma_{m}V$ must fill everything. ∎

We now state an important result by Massarenti and Mella [18], which enables
us to link the study of defectivity of varieties $V$ to their identifiability
properties.

######  3.6 Theorem (Massarenti, Mella, [18, Theorem 1.5]).

Let $V\subseteq\mathcal{L}$ be an irreducible and non-degenerate variety of
dimension $N$, $m\geq 1$ an integer and assume that:

  1. (1)

$mN\leq\dim\mathcal{L}$,

  2. (2)

The fibers of the tangent map
$T\colon\mathbb{P}(V_{\operatorname{reg}})\to\operatorname{Gr}(N,\mathcal{L})$
are finite.

  3. (3)

$V$ is not $(m+1)$-defective.

Then, $V$ is $m$-identifiable.

If a variety $V$ satisfies Condition (2) of Theorem 3.6, we also say that the
tangent map of $V$ is _nondegenerate_.

##  4\. Applications

### Grassmannian varieties

######  4.1 Notation.

Let $d,n\in\mathbb{N}$. We denote by $\operatorname{Gr}(d,n)$ the
_Grassmannian variety_ of $d$-dimensional subspaces of $\mathbb{C}^{n}$. Via
the _Plücker embedding_ , we may see $\operatorname{Gr}(d,n)$ as a subvariety
of the Schur functor $\bigwedge^{d}(\mathbb{C}^{n})$.

Secants to Grassmannian varieties are a natural object to study in tensor
geometry. Just a few years after the celebrated Alexander-Hirschowitz result,
Catalisano, Geramita and Gimigliano examined their nondefectivity [9]. Galgano
[13] examined non-generic identifiability for
$\sigma_{2}\operatorname{Gr}(d,n)$. Baur, Draisma and de Graaf [16] discovered
connections between secant nondefectivity of Grassmannians and coding theory.
In addition, their work lead to the following conjecture.

######  4.2 Conjecture (Baur–Draisma–de Graaf).

Let $n,d,m\in\mathbb{N}$ and $d\geq 3$. Then, the secant variety
$\sigma_{m}\operatorname{Gr}(d,n)$ has the expected dimension $\min(m\cdot
d(n-d),\binom{n}{d})$, except for the following cases:
Actual Codimension Expected Codimension $\sigma_{3}\operatorname{Gr}(3,7)$ 1 0
$\sigma_{3}\operatorname{Gr}(4,8)$ 20 19 $\sigma_{4}\operatorname{Gr}(4,8)$ 6
2 $\sigma_{4}\operatorname{Gr}(3,9)$ 10 8

The table in 4.2 was taken from [5, Conj 4.2]. Note that we adapted to affine
notation, so in our table, both $d$ and $n$ are each larger by $1$ compared to
the table in [5, Conj 4.2]. If it was known to be true, 4.2 could be seen as
an “alternating” version of the Alexander-Hirschowitz theorem, where the Schur
functor $S^{d}$ of symmetric powers is replaced by the Schur functor
$\bigwedge^{d}$ of alternating powers, and the Veronese embedding is replaced
by the Plücker embedding. Fortunately, with 3.5, we are able to resolve a lot
of cases of the Baur-Draisma-de Graaf conjecture.

######  4.3 Theorem.

For any $d,n,m\in\mathbb{N}$, the Grassmannian variety
$\operatorname{Gr}(d,n)$ is $m$-nondefective (and $(m-1)$-identifiable), if

$\displaystyle m$
$\displaystyle\leq\frac{\dim\bigwedge^{d}(\mathbb{C}^{n})}{\dim\operatorname{Gr}(d,n)}-\dim\operatorname{Gr}(d,n)$
(25) $\displaystyle=\frac{1}{d(n-d)}\binom{n}{d}-d(n-d)$ (26)

Conversely, the generic rank of $\bigwedge^{d}(\mathbb{C}^{n})$ with respect
to $\operatorname{Gr}(d,n)$ is at most

$\displaystyle\frac{1}{d(n-d)}\binom{n}{d}+d(n-d).$ (27)

###### Proof.

Since the Grassmannian is a smooth variety, its tangent map is nondegenerate.
Therefore, $m$-nondefectivity implies $(m-1)$-identifiability, by Theorem 3.6.
The rest follows from 3.5. ∎

######  4.4 Remark.

Our Theorem 4.3 settles a large number of cases the Baur–Draisma–de Graaf
conjecture stated below. In fact, for any $d\geq 3$, the number of ranks $m$
for which the conjecture is not settled by our Theorem 4.3 is asymptotically
negligible as a function in $n$, for large $n$. For $d\leq 2$, our theorem is
vacuous. This is also necessary, since secants to Grassmannians of two-
dimensional subspaces are known to be non-identifiable.

Let us compare our Theorem 4.3 with earlier results. Catalisano, Geramita and
Gimigliano [9] showed nondefectivity for all $m\leq n/d$, if $d\geq 3$.
Rischter [23, Theorem 3.5.1] showed nondefectivity for all $m$ bounded by some
complicated function in $n$ and $d$, with asymptotic growth around
$\mathcal{O}(n^{\log_{2}(d)+1})$. Both results are collected in the
“hitchhiker guide” of Bernardi et. al., see [5, Theorem 4.3] and [5, Theorem
4.5]. For large $n$, our bound is of course strictly better than both previous
results, and it matches the asymptotics of the Baur-Draisma-de Graaf
conjecture. Figure 1 shows that it already overtakes the bounds from [23] and
[9] in the case $d=4$ at $n=20$.

Figure 1. Comparison of the various bounds on secant nondefectivity of
$\operatorname{Gr}(d,n)$ when $d=5$.

From Theorem 1.1, similar identifiability results may also be derived for flag
varieties.

######  4.5 Notation.

Let $(k_{1},\dots,k_{r};n)$ denote a sequence of increasing positive numbers
of the form $k_{1}\leq\dots\leq k_{r}\leq n$. We denote by
$F(k_{1},\dots,k_{r};n)$ the _Flag variety_ parametrizing chains of the forms

$V_{1}\subset\dots\subset V_{r}\subset\mathbb{C}^{n}$

where every $V_{i}$ is a linear subspace of dimension $\dim(V_{i})=k_{i}$.

In particular the Flag variety can be seen as a subvariety of the product of
Grassmannians $\prod_{i=1}^{r}\operatorname{Gr}(k_{i},n)$. Restricting the
Segre product of the Plücker embedding to $F(k_{1},\dots,k_{r};n)$ exhibits
the Flag as a subvariety of the irreducible $\operatorname{GL}_{n}^{r}$-module
$\Gamma_{a}\subseteq\bigwedge^{k_{1}}\mathbb{C}^{n}\otimes\ldots\otimes\bigwedge^{k_{r}}\mathbb{C}^{n}$,
see for instance [10]. Note that, if we define $k_{0}:=0$, we have

$\dim F(k_{1},\dots,k_{r};n)=\sum_{j=1}^{r}(n-k_{j})(k_{j}-k_{j-1})$

To the best of our knowledge the best bound for secant defectivity and
identifiability for Flag varieties is given in [10, Theorem 4.11], where the
authors proved that under the assumption $n\geq 2k_{r}$ the Flag variety
$F(k_{1},\dots,k_{r};n)$ is not $m$-defective for

$\displaystyle m\leq\left(\frac{n}{k_{r}}\right)^{\lfloor\log_{2}(\sum
k_{j}-1)\rfloor}$ (28)

Using our result from Theorem 1.1 we can give the following improvement:

######  4.6 Theorem.

The Flag variety $F(k_{1},\dots,k_{r};n)$ is $m$-nondefective (and
$(m-1)$-identifiable), if $m$ is bounded by:

$\displaystyle\frac{\prod_{i=1}^{r}{n\choose
k_{i}}}{\sum_{j=1}^{r}(n-k_{j})(k_{j}-k_{j-1})}-\sum_{j=1}^{r}(n-k_{j})(k_{j}-k_{j-1})$
(29)

Furthermore, the generic rank $m^{\circ}$ is at most

$\displaystyle\frac{\prod_{i=1}^{r}{n\choose
k_{i}}}{\sum_{j=1}^{r}(n-k_{j})(k_{j}-k_{j-1})}+\sum_{j=1}^{r}(n-k_{j})(k_{j}-k_{j-1})$
(30)

### Chow varieties

In recent work by Torrance and Vannieuwenhoven, see [26], [25], it was shown
that almost all secants of the Chow variety

$\displaystyle\operatorname{Chow}_{d}(\mathbb{C}^{n})=\\{\ell_{1}\cdot\ldots\cdot\ell_{d}\mid\ell_{1},\ldots,\ell_{d}\in
S^{1}(\mathbb{C}^{n})\\}$ (31)

are identifiable, in the case $d=3$ of cubics. We can show a result for all
degrees $d\geq 3$.

######  4.7 Theorem.

The Chow variety $\operatorname{Chow}_{d}(\mathbb{C}^{n})$ is $m$-nondefective
(and $(m-1)$-identifiable), if $m$ is at most
$\frac{1}{d(n-1)+1}\binom{n+d-1}{d}-d(n-1)-1$. Its generic rank is at most
$\frac{1}{d(n-1)+1}\binom{n+d-1}{d}+d(n-1)+1$.

###### Proof.

The Chow variety is $\operatorname{GL}$-invariant and lives in the Schur
functor $S^{d}(\mathbb{C}^{n})$. The statement thus follows from Theorem 1.1,
after plugging in the dimension of $\operatorname{Chow}_{d}(\mathbb{C}^{n})$,
which is $(n-1)d+1$, and of $S^{d}(\mathbb{C}^{n})$. Oeding computed the
dimension of the dual of the Chow variety in [20]. As Torrance and
Vannieuwenhoven pointed out in [25, Section 2.1], Oeding’s result implies that
$\operatorname{Chow}_{d}(\mathbb{C}^{n})$ is not $1$-weakly defective for
$d\geq 3$ and $n\geq 2$. Hence, the Chow variety is also not $1$-tangentially
weakly defective and Theorem 3.6 implies $(m-1)$-identifiability. ∎

### Segre-Veronese varieties

Given $t\in\mathbb{N}$, vector spaces $\underline{L}=(L_{1},\ldots,L_{t})$ of
dimensions $\underline{n}=(n_{1},\ldots,n_{t})$ and nonnegative integers
$\underline{d}=(d_{1},\ldots,d_{t})$, the Segre-Veronese variety on
$\underline{L}$ of multidegree $\underline{d}$ is defined as

$\displaystyle\operatorname{s\nu}_{\underline{d}}(\underline{L}):=\\{\ell_{1}^{d_{1}}\otimes\ldots\otimes\ell_{t}^{d_{t}}\mid\ell_{1}\in
L_{1}^{\vee},\ldots,\ell_{t}\in L_{t}^{\vee}\\}.$ (32)

It lives in $S^{d_{1}}(L_{1})\otimes\ldots\otimes S^{d_{t}}(L_{t})$. This
space is naturally endowed with the action of
$G:=\operatorname{GL}(L_{1})\times\ldots\times\operatorname{GL}(L_{t})$.

######  4.8 Theorem.

The Segre-Veronese variety of multidegree $\underline{d}$ and dimensions
$\underline{n}$ is $m$-nondefective (and $(m-1)$-identifiable), if $m$ is
bounded by

$\displaystyle\frac{1}{n-t+1}\binom{n_{1}+d_{1}-1}{d_{1}}\cdots\binom{n_{t}+d_{t}-1}{d_{t}}-(n-t+1),$
(33)

where $n:=n_{1}+\ldots+n_{t}$. Furthermore, the generic rank $m^{\circ}$ is at
most

$\displaystyle\frac{1}{n-t+1}\binom{n_{1}+d_{1}-1}{d_{1}}\cdots\binom{n_{t}+d_{t}-1}{d_{t}}+(n-t+1).$
(34)

###### Proof.

As outlined before, $\mathcal{L}:=S^{d_{1}}(L_{1})\otimes\ldots\otimes
S^{d_{t}}(L_{t})$ has the structure of a $G$-module, where
$G:=\operatorname{GL}(L_{1})\times\ldots\times\operatorname{GL}(L_{t})$. But
$\mathcal{L}$ is in fact an irreducible $G$-module, since it is the tensor
product of the irreducible $\operatorname{GL}(L_{i})$-modules
$S^{d_{i}}(L_{i})$, for $i\in\\{1,\ldots,t\\}$. Note that
$\operatorname{s\nu}_{\underline{d}}(\underline{L})$ is invariant with respect
of the action of $G$, and of dimension $n_{1}+\ldots+n_{t}-t+1$. Therefore,
Theorem 1.1 is applicable. Calculating the dimensions of $\mathcal{L}$ and the
Segre-Veronese variety explicitly yields the result on nondefectivity. Since
Segre-Veronese varieties are smooth, Theorem 3.6 yields
$(m-1)$-identifiability. ∎

To the best of our knowledge the actual best bound in the literature is due to
Laface-Massarenti-Rischter in [17, Theorem 1.1]. They give an asymptotical
sharp bound on $m$ of the form:

$\displaystyle\frac{d_{j}}{(n_{j}+d_{j}-1)\cdot(n-t+1)}\binom{n_{1}+d_{1}-1}{d_{1}}\cdots\binom{n_{t}+d_{t}-1}{d_{t}}$
(35)

where

$\frac{n_{j}-1}{d_{j}}=\max_{1\leq i\leq t}\\{\frac{n_{i}-1}{d_{i}}\\}.$

Our bound from Theorem 4.8 however covers asymptotically more cases then
Laface-Massarenti-Rischter one. In Figure 2 we make the comparison between the
two bounds in the case of $3$ factors Segre-Veronese of fixed degree
$\underline{d}=(1,1,2)$ and fixed dimension $n$.

Figure 2. Comparison of the various bounds on secant nondefectivity of
$s\nu_{\underline{d}}(\underline{n})$ when $\underline{d}=(1,1,2)$ and
$\underline{n}=(n,n,n)$.

### The Gaussian moment variety

Gaussian mixtures are an ubiquitous model in Machine Learning. Their
identifiability is a long-standing open problem, with the univariate case
dating back to Pearson, see [21]. We refer to [2] and [3] for some historical
remarks. They are closely connected to secants of the _Gaussian moment
variety_ , defined as follows.

######  4.9 Definition.

Let $n,d\in\mathbb{N}$. We call the closure of the image of

$\displaystyle s_{d}\colon S^{1}(\mathbb{C}^{n})\times
S^{2}(\mathbb{C}^{n})\to
S^{d}(\mathbb{C}^{n}),(\ell,q)\mapsto\exp\left(\ell+\frac{q}{2}\right)_{=d},$
(36)

the degree-$d$ _Gaussian moment variety_
$\operatorname{GM}_{d}(\mathbb{C}^{n})$ on $\mathbb{C}^{n}$. Here, $\exp$ is
to be understood as a formal power series on $\mathbb{C}^{n}$ and the
subscript denotes its $d$-homogeneous part.

The parameter $\ell$ is a linear form, which corresponds to the mean of the
Gaussian, while the parameter $q$ is a quadratic form, corresponding to the
covariance matrix. In [7] and [6], the authors gave thorough motivations of
the Gaussian moment variety and explained the connection between the
statistical estimation problem for Gaussian mixtures and the algebraic
identifiability problem for the Gaussian moment variety. In particular, they
also showed why for generic identifiability, the constrains of realness and
positive-definiteness of $\ell$ and $q$, respectively, may be ignored. We
settle most cases of Gaussian identifiability for high-dimensional models in
the degrees $d=5,\ldots,9$. Note that in applications, the lower degrees are
the most relevant ones. Degree $5$ is minimal for any nontrivial
identifiability result.

######  4.10 Theorem.

For $d\geq 5$ and $n\geq 2$, the following hold:

  1. (1)

The Gaussian moment varieties $\mathrm{GM}_{d}(\mathbb{C}^{n})$ are
$m$-nondefective, as long as

$\displaystyle m\leq\frac{\dim
S^{d}(\mathbb{C}^{n})}{\dim\mathrm{GM}_{d}(\mathbb{C}^{n})}-\dim\mathrm{GM}_{d}(\mathbb{C}^{n})=\frac{2{n+d-1\choose
d}}{n(n+3)}-\frac{1}{2}n(n+3).$

  2. (2)

In addition, the generic rank with respect to the Gaussian moment variety is
at most

$\displaystyle\frac{{n+d-1\choose
d}}{\dim\mathrm{GM}_{d}(\mathbb{C}^{n})}+\dim\mathrm{GM}_{d}(\mathbb{C}^{n}).$

  3. (3)

For $d=5,\ldots,9$, $m$-nondefectivity of $\mathrm{GM}_{d}(\mathbb{C}^{n})$
implies $(m-1)$-identifiability.

###### Proof.

The first two claims are a direct consequence of Theorem 1.1. Note that the
dimension of $\operatorname{GM}_{d}(\mathbb{C}^{n})$ is $n+\binom{n+1}{2}$,
since it is parameterized by $S^{1}(\mathbb{C}^{n})\oplus
S^{2}(\mathbb{C}^{n})$. This value can also be written as $\frac{1}{2}n(n+3)$.
For the third claim, we use Theorem 3.6, which shows that $m$-nondefectivity
implies $(m-1)$-identifiability, if the tangent map is nondegenerate. The
tangent map condition was verified for $d=5,\ldots,9$ with a partially
computer-assisted proof in [6, Proposition 2.13]. ∎

######  4.11 Remark.

For $d\leq 4$, the Gaussian moment varieties are already $2$-defective, cf.
for instance [6]. On the other hand, for $d\geq 5$, the dimension of
$\mathrm{GM}_{d}(\mathbb{C}^{n})$ grows of smaller order in $n$ than the
parameter counting rank. Therefore, Theorem 4.10 asymptotically covers almost
all interesting cases of nondefectivity, cf. Remark 4.17. We conjecture that
$\mathrm{GM}_{d}(\mathbb{C}^{n})$ has nondegenerate tangent map for all $d\geq
5$ and $n\geq 2$. A proof of this would extend Theorem 4.10(3) to all degrees
$d\geq 5$.

#### Low-rank Gaussians

We can prove a similar result for rank-$r$ Gaussians.

######  4.12 Corollary.

Let $n,m,d,r\in\mathbb{N}$ and denote

$\displaystyle\operatorname{GM}_{d,r}(\mathbb{C}^{n})=\\{s_{d}(\ell,q)\mid\operatorname{rank}q\leq
r\\}.$ (37)

Then, $\operatorname{GM}_{d,r}(\mathbb{C}^{n})$ is $m$-nondefective, as long
as

$\displaystyle m\leq\frac{\dim S^{d}(\mathbb{C}^{n})}{M_{r}}-M_{r},$ (38)

where $M_{r}:=(r+1)n-\binom{r}{2}$. The generic rank is at most $\frac{\dim
S^{d}(\mathbb{C}^{n})}{M_{r}}+M_{r}$.

###### Proof.

Counting parameters, we have that for $d\geq 3$,
$\operatorname{GM}_{d}(\mathcal{G}_{r})$ has dimension equal to $n$ (for the
linear form) plus $r(n-r)+\binom{r+1}{2}=rn-\binom{r}{2}$ for the rank-$r$
covariance form. This sum equals $M_{r}=(r+1)n-\binom{r}{2}$. Clearly,
$\operatorname{GM}_{d}(\mathcal{G}_{r})$ is a
$\operatorname{GL}(\mathbb{C}^{n})$-invariant, irreducible variety. The claim
thus follows by 3.5. ∎

######  4.13 Remark.

4.12 treats one example in the large category of _Gaussian models_. These
models are very common in applications and they arise from considering some
subvariety $\mathcal{G}\subseteq S^{2}(\mathbb{C}^{n})$. The major motivation
for a restriction to subvarieties is to reduce the number of parameters. This
parameter reduction can have a significant impact on defectivity: Indeed, 4.12
implies that, if the rank of the covariance matrices is an absolute constant,
then a mixture of $m=\mathcal{O}(n^{2})$ general Gaussians is finite-to-one
identifiable from its degree $3$ moments; a sharp contrast to the case of full
Gaussians, where moments of degree $5$ are necessary. We believe that
nondefectivity results for the rank-$r$ Gaussian mixture model were not known
before.

### Other $G$-varieties

In this section, we will see two applications of Theorem 1.1 for the case of
other groups. Let $\mathbb{C}^{2n}$ be a $2n$-dimensional complex vector space
and let $Q$ and $\omega$, respectively, a non-degenerate quadratic form and a
non-degenerate skew symmetric form. Let $G=\operatorname{SO}(\mathbb{C}^{2n})$
be the special orthogonal group with respect to $Q$ and let
$G=\operatorname{Sp}(\mathbb{C}^{2n})$ denote the symplectic group with
respect to $\omega$. We briefly recall the notations and the main result from
[11].

######  4.14 Definition.

A subspace $L\subseteq\mathbb{C}^{2n}$ of dimension $n$ is called
$p$-_isotropic_ with respect to the nondegenerate bilinear form $p$, if for
all $x\in L$, $p(x,x)=0$.

We now introduce two special subvarieties of the Grassmannian that are
invariants under the two groups introduced above.

######  4.15 Definition.

  1. (a)

The _Lagrangian Grassmannian_
$\mathcal{LG}(n,2n)\subset\operatorname{Gr}(n,2n)$ is the variety of
$n$-dimensional $\omega$-isotropic linear subspaces of $\mathbb{C}^{2n}$. It
is $\operatorname{Sp}(\mathbb{C}^{2n})$-invariant and has dimension
$\frac{n(n+1)}{2}$.

  2. (b)

The _Spinor variety_ $\mathcal{S}_{n}\subset\operatorname{Gr}(n,2n)$ is the
variety of $n$-dimensional $Q$-isotropic linear subsapces of
$\mathbb{C}^{2n}$. It is $\operatorname{SO}(\mathbb{C}^{2n})$-invariant of
dimension $\frac{n(n-1)}{2}$.

The restriction of the Plücker map to $\mathcal{LG}(n,2n)$ embeds the
Lagrangian Grassmannian in the irreducible
$\operatorname{Sp}(\mathbb{C}^{2n})$-module $\mathbb{C}^{2n}_{\omega}$. The
latter is defined as the complement of the submodule
$\omega\wedge\bigwedge^{n-2}\mathbb{C}^{2n}\subset\bigwedge^{n}\mathbb{C}^{2n}$.
It holds

$\displaystyle\dim\mathbb{C}^{2n}_{\omega}=\frac{1}{2}\binom{2n}{n}+2^{n-1}-1$
(39)

Recall that $\mathcal{S}_{n}$ has two connected isomorphic components
$\mathcal{S}_{n}^{+}$ and $\mathcal{S}_{n}^{-}$. Here, we implicitely make the
choice $\mathcal{S}_{n}=\mathcal{S}_{n}^{+}$. Note that the restriction of the
Plücker map embeds the Spinor variety into an irreducible
$\operatorname{SO}(\mathbb{C}^{2n})$-module
$\bigwedge^{n}\mathbb{C}^{2n}_{+}$, where
$\bigwedge^{n}\mathbb{C}^{2n}_{+}\oplus\bigwedge^{n}\mathbb{C}^{2n}_{-}=\bigwedge^{n}\mathbb{C}^{2n}$
is the natural $\operatorname{SO}(\mathbb{C}^{2n})$-decomposition. The
embedding $\mathcal{S}_{n}\to\bigwedge^{n}\mathbb{C}^{2n}_{+}$ is not the
minimal one: Indeed, there is another embedding $\mathcal{S}_{n}\to\Delta$,
where $\Delta$ is the half-spin $\operatorname{SO}(\mathbb{C}^{2n})$
representation. We have

$\displaystyle\dim\bigwedge^{n}\mathbb{C}^{2n}_{+}=\frac{1}{2}{2n\choose
n}\text{ and }\dim\Delta=2^{n-1}$ (40)

The authors in [11] proved that
$\mathcal{LG}(n,2n)\subset\mathbb{C}^{2n}_{\omega}$ is not $m$-defective for
$m$ at most $\lfloor\frac{n+1}{2}\rfloor$. Moreover, they showed that
$\mathcal{S}_{n}\subset\bigwedge^{n}\mathbb{C}^{2n}_{+}$ is not $m$-defective
for $m\leq\lfloor\frac{n}{2}\rfloor$ and that $\mathcal{S}_{n}\subset\Delta$
is not $m$-defective for $m\leq\lfloor\frac{n+2}{4}\rfloor$. Using Theorem
1.1, we are able to significantly improve on the bounds of Massarenti and
Rischter:

######  4.16 Theorem.

The Lagrangian Grassmannian $\mathcal{LG}(n,2n)$ is $m$-nondefective (and
$(m-1)$-identifiable) in the $\mathbb{C}^{2n}_{\omega}$ embedding, if $m$ is
bounded by

$\displaystyle\frac{1}{n(n+1)}\left(\binom{2n}{n}+2^{n}-2\right)-\frac{n(n+1)}{2}$
(41)

Furthermore, the generic rank $m^{\circ}$ is at most

$\displaystyle\frac{1}{n(n+1)}\left(\binom{2n}{n}+2^{n}-2\right)+\frac{n(n+1)}{2}$
(42)

Similarly, the Spinor variety $\mathcal{S}_{n}$ is $m$-nondefective (and
$(m-1)$-identifiable) in the $\bigwedge^{n}\mathbb{C}^{2n}_{+}$ embedding, if
$m$ is bounded by

$\displaystyle\frac{1}{n(n-1)}{2n\choose n}-\frac{n(n-1)}{2}$ (43)

and in the $\Delta$ embedding, if $m$ is bounded by

$\displaystyle\frac{2^{n}}{n(n-1)}-\frac{n(n-1)}{2}$ (44)

An analogous statement follows for the generic rank $m^{\circ}$.

Identifiability follows again from [18, Theorem 1.5], since both spinor
varieties and Lagrangian Grassmannians are smooth. Galgano completely
determined the locus of identifiability for the second secant of the spinor
variety [12].

### Asymptotics and critical degree

######  4.17 Remark.

The examples from Section 1 will be discussed thoroughly in Section 4.
However, note that the examples from the introduction have slightly more in
common than just $\operatorname{GL}$-invariance: Veronese varieties, powers-
of-forms varieties, Chow varieties, Grassmann varieties and Gaussian moment
varieties are images $\varphi_{d}(L_{d})$ (under equivariant maps) of a linear
space $L_{d}=L_{d}(\mathbb{C}^{n})$ into spaces
$\mathcal{L}_{d}(\mathbb{C}^{n})$ of $d$-tensors on $\mathbb{C}^{n}$.111In
fact, $L_{d}(\mathbb{C}^{n})$ has the structure of a bounded-degree polynomial
functor, for fixed $d$.  In all of the above cases, the dimension of
$L_{d}(\mathbb{C}^{n})$ is a polynomial $p_{d}(n)$ in $n$, whose degree
$d_{0}:=\deg p_{d}$ does not depend on $d$.

Concretely, we have $d_{0}=1$ for the Veronese, Chow and Grassmannian
varieties. We have $d_{0}=2$ for the Gaussian moment variety and $d_{0}=k$ for
the variety of $d$-th powers of $k$-forms. We may now look at the bounds in
Theorem 1.1 and understand their asymptotical behaviour for $n\to\infty$, just
in terms of $d_{0}$. Indeed, if $d>2d_{0}$, then for large $n$, almost all
secants of $\varphi_{d}(L_{d}(\mathbb{C}^{n}))$ will be identifiable. The
number of cases not covered by our theorem will be asymptotically negligible.
The generic rank will be very close to the parameter count
$\frac{\dim\mathcal{L}_{d}(\mathbb{C}^{n})}{\dim\varphi_{d}(L_{d}(\mathbb{C}^{n}))}$.

On the other hand, for $d<2d_{0}$, our theorem becomes trivial (at least for
large $n$, but in practice often really fast). This is necessary, since in all
of the examples above, for $d\leq d_{0}$, already the second secant is
defective. Summarized, it can be said that all of the above examples have a “
_critical degree_ ” $2d_{0}$, which separates the $\varphi_{d}$ with mostly
defective secants from those with mostly nondefective secants.

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# A Non-Iterative Transformation Method for an Extended Blasius Problem

Riccardo Fazio
Department of Mathematics, Computer Science
Physical Sciences and Earth Sciences,
University of Messina
Viale F. Stagno D’Alcontres, 31
98166 Messina, Italy
E-mail<EMAIL_ADDRESS>
Home-page: http://mat521.unime.it/fazio

###### Abstract

In this paper, we define a non-iterative transformation method for an Extended
Blasius Problem. The original non-iterative transformation method, which is
based on scaling invariance properties, was defined for the classical Blasius
problem by Töpfer in 1912. This method allows us to solve numerically a
boundary value problem by solving a related initial value problem and then
rescaling the obtained numerical solution. In recent years, we have seen
applications of the non-iterative transformation method to several problems of
interest.

The obtained numerical results are improved by both a mesh refinement strategy
and Richardson’s extrapolation technique. In this way, we can be confident
that the computed six decimal places are correct.

Key Words. Extended Blasius problem; scaling invariance properties; non-
iterative transformation method; BVPs on infinite intervals.

AMS Subject Classifications. 65L10, 34B15, 65L08.

##  1 Introduction.

The problem of determining the steady two-dimensional motion of a fluid past a
flat plate placed edge-ways to the stream was formulated in general terms,
according to the boundary layer theory, by Prandtl [32], and was investigated
in detail by Blasius [2]. The engineering interest was to calculate the shear
at the plate (skin friction), which leads to the determination of the viscous
drag on the plate, see for instance Schlichting [38]). In this contest the
celebrated Blasius problem is given by

$\displaystyle{\displaystyle\frac{d^{3}f}{d\eta^{3}}}+\frac{1}{2}\;f{\displaystyle\frac{d^{2}f}{d\eta^{2}}}=0$
$\displaystyle
f(0)={\displaystyle\frac{df}{d\eta}}(0)=0,\qquad{\displaystyle\frac{df}{d\eta}}(\eta)\rightarrow
1\quad\mbox{as}\quad\eta\rightarrow\infty\ .$

This is a boundary value problem (BVP) defined on the semi-infinite interval
$[0,\infty)$. According to Weyl [41], the unique solution of (1) has a
positive second order derivative, which is monotone decreasing on $[0,\infty)$
and approaches to zero as $\eta$ goes to infinity. The governing differential
equation and the two boundary conditions at the origin in (1) are invariant
with respect to the scaling group of transformations

$\displaystyle\eta^{*}=\lambda^{-\alpha},\qquad f^{*}=\lambda^{\alpha}f$ (2)

where $\alpha$ is a nonzero constant, classically $\alpha=1/3$; but here we
have used also $\alpha=1$ in order to simplify the analysis. The mentioned
invariance property has both analytical and numerical interest. From a
numerical viewpoint a non-iterative transformation method (ITM) reducing the
solution of (1) to the solution of a related initial value problem (IVP) was
defined by Töpfer [40]. Owing to that transformation, a simple existence and
uniqueness Theorem was given by J. Serrin as reported by Meyer [29, pp.
104-105]. Let us note here that the mentioned invariance property is essential
to the error analysis of the truncated boundary solution due to Rubel [35],
see Fazio [13]. Blasius problem was used, recently, by Boyd [4] as an example
were some good analysis, before the computer invention, allowed researchers of
the past to solve problems, governed by partial differential equations, that
might be otherwise impossible to face.

The application of a non-ITM to the Blasius equation with slip boundary
condition, arising within the study of gas and liquid flows at the micro-scale
regime [5, 28], was considered already in [14]. We applied a non-ITM to the
Blasius equation with moving wall considered by Ishak et al. [21] or surface
gasification studied by Emmons [6] and recently by Lu and Law [27] or slip
boundary conditions investigated by Gad-el-Hak [5] or Martin and Boyd [28],
see Fazio [16] for details. In particular, we find a way to solve non-
iteratively the Sakiadis problem [36, 37]. As far as the non-ITM is concerned,
a recent review dealing with all the cited problems can be be found in [18].

Moreover, Töpfer’s method has been extended to classes of problems in boundary
layer theory involving a physical parameter. This kind of extension was
considered first by Na [30], see also NA [31, Chapters 8-9].

Finally, an iterative extension of the transformation method has been
introduced, for the numerical solution of free BVPs, by Fazio [19]. This
iterative extension has been applied to several problems of interest: free
boundary problems [19, 10, 11], a moving boundary hyperbolic problem [8],
Homann and Hiemenz problems governed by the Falkner-Skan equation in [9], one-
dimensional parabolic moving boundary problems [12], two variants of the
Blasius problem [14], namely: a boundary layer problem over moving surfaces,
studied first by Klemp and Acrivos [23], and a boundary layer problem with
slip boundary condition, that has found application in the study of gas and
liquid flows at the micro-scale regime [5, 28], parabolic problems on
unbounded domains [20] and, recently, see [15], a further variant of the
Blasius problem in boundary layer theory: the so-called Sakiadis problem [36,
37]. A recent review dealing with, the derivation and application of, ITM can
be be found by the interested reader in [17].

##  2 Extended Blasius problem

Our extended Blasius problem is given by,

$\displaystyle{\displaystyle\frac{d^{3}f}{d\eta^{3}}}{\displaystyle\frac{d^{2}f}{d\eta^{2}}}^{(P-1)}+\frac{1}{2}\;f{\displaystyle\frac{d^{2}f}{d\eta^{2}}}=0$
$\displaystyle
f(0)={\displaystyle\frac{df}{d\eta}}(0)=0,\qquad{\displaystyle\frac{df}{d\eta}}(\eta)\rightarrow
1\quad\mbox{as}\quad\eta\rightarrow\infty\ ,$

where $P$ verifies the conditions $1\leq P<2$, see Schowalter [39], Lee and
Ames [24], Lin and Chern [26], Kim et al. [22], or Akcay and Yükselen [1].
Liao [25] has found analytically that the extended Blasius problem for $P=2$
admit an infinite number of solutions anf therefore in his opinion can be
considered as a challenge problem for numerical techniques.

###  2.1 The non-ITM

The applicability of a non-ITM to the Blasius problem (1) is a consequence of
both: the invariance of the governing differential equation and the two
boundary conditions at $\eta=0$, and the non invariance of the asymptotic
boundary condition under the scaling transformation (2). In order to apply a
non-ITM to the BVP (2) we investigate its invariance with respect to the
scaling group

$f^{*}=\lambda f\ ,\qquad\eta^{*}=\lambda^{\delta}\eta\ .$ (4)

We find that the extended Blasius problem (2) is invariant under (4) iff

$\delta=\frac{2-P}{1-2P}\ .$ (5)

Now, we can integrate the extended Blasius equation in (2) written in the star
variables on $[0,\eta^{*}_{\infty}]$, where $\eta^{*}_{\infty}$ is a suitable
truncated boundary, with initial conditions

$f^{*}(0)=\frac{df^{*}}{d\eta^{*}}(0)=0\
,\quad\frac{d^{2}f^{*}}{d\eta^{*2}}(0)=1\ ,$ (6)

in order to compute an approximation
$\frac{df^{*}}{d\eta^{*}}(\eta^{*}_{\infty})$ for
$\frac{df^{*}}{d\eta^{*}}(\infty)$ and the corresponding value of $\lambda$
according to the equation

$\lambda=\left[\frac{df^{*}}{d\eta^{*}}(\eta^{*}_{\infty})\right]^{1/(1-\delta)}\
.$ (7)

Once the value of $\lambda$ has been computed by equation (7), we can find the
missed initial condition by the equation

$\frac{d^{2}f}{d\eta^{2}}(0)=\lambda^{2\delta-1}\frac{d^{2}f^{*}}{d\eta^{*2}}(0)\
.$ (8)

Moreover, the numerical solution of the original BVP (2) can be computed by
rescaling the solution of the IVP. In this way we get the solution of a given
BVP by solving a related IVP.

##  3 Numerical results

In this section we report the numerical results computed with our non-ITM. To
compute the numerical solution, we used the classical fourth order Runge-Kutta
method with constant step size. For the results shown in this figure we used
$\Delta\eta=0.001$ and $\eta^{*}_{\infty}=5$. Figure 1 shows the solution of
the extended Blasius problem, describing the behaviour of a boundary layer
flow due to a moving flat surface immersed in an otherwise quiescent fluid,
corresponding to $P=3/2$.

Figure 1: Numerical results of the non-ITM for (2) with $P=3/2$. The starred
variables problem and the original problem solution components found after
rescaling.

Let us notice here that, by rescaling, we get
$\eta^{*}_{\infty}<\eta_{\infty}$.

In table 1 we report the missing initial condition
$\frac{d^{2}f}{d\eta^{2}}(0)$ obtained by a mesh refinement.

Table 1: Missing initial condition by mesh refinement. ${\Delta\eta}$ | $\frac{d^{2}f}{d\eta^{2}}(0)$
---|---
0.001 | 0. | 47001559210
0.0005 | 0. | 46953524356
0.00025 | 0. | 46929513793
0.000125 | 0. | 46917509666
0.0000625 | 0. | 46911507775
0.00003125 | 0. | 46908506833
0.000015625 | 0. | 46907006359
0.0000078125 | 0. | 46906256120

As it is easily seen the decimal digits, starting to the left of the decimal
point, tend to be confirmed, in particular, here we have confirmed the first
four decimal values.

In order to improve the obtained missing initial condition value, we can apply
a Richardson’s extrapolation technique, see [33, 34]. Let us consider that we
would like to compute the value $U=\frac{d^{2}f}{d\eta^{2}}(0)$ as
$\Delta\eta$ goes to zero, then we can apply the extrapolation formula

$U_{g+1,k+1}=U_{g+1,k}+\frac{U_{g+1,k}-U_{g,k}}{4-1}\ ,$ (9)

where $g\in\\{0,1,2,\dots,G-1\\}$, $k\in\\{0,1,2,\dots,G-1\\}$, here $g$
indicates the $g$-esima computational grid, $k$ the level of extrapolation,
and $4$ is the true order of the discretization error for the classical Runge-
Kutta method. The formula (9) is asymptotically exact in the limit as
$\Delta\eta$ goes to zero if we use uniform grids. We notice that to obtain
each value of $U_{g+1,k+1}$ requires two computed solutions $U$ in two
adjacent grids, namely $g+1$ and $g$ at the extrapolation level $k$. For any
$g$, the level $k=0$ represents the numerical solution of $U$ without any
extrapolation. The numerical results obtained applying formula (9) to the
values reported in table 1 are reported in table 2.

Table 2: Missing initial condition by Richardson’s extrapolation. $U_{g,0}$ | $U_{g,1}$ | $U_{g,2}$ | $U_{g,3}$ | $U_{g,4}$ | $U_{g,5}$ | $U_{g,6}$ | $U_{g,7}$
---|---|---|---|---|---|---|---
0.47001559210 |  |  |  |  |  |  |
0.46953524356 | 0.46937512738 |  |  |  |  |  |
0.46929513793 | 0.46921510272 | 0.46916176117 |  |  |  |  |
0.46917509666 | 0.46913508290 | 0.46910840963 | 0.46909062578 |  |  |  |
0.46911507775 | 0.46909507145 | 0.46908173430 | 0.46907284252 | 0.46906691477 |  |  |
0.46908506833 | 0.46907506519 | 0.46906839644 | 0.46906395049 | 0.46906098648 | 0.46905901038 |  |
0.46907006359 | 0.46906506201 | 0.46906172762 | 0.46905950468 | 0.46905802274 | 0.46905703482 | 0.46905637630 |
0.46906256120 | 0.46906006040 | 0.46905839320 | 0.46905728173 | 0.46905654074 | 0.46905604674 | 0.46905571739 | 0.46905549774

Naturally, we can stop the mesh refinement as soon as $U_{g+1,k}=U_{g,k}$ or
Richardon’s extrapolation when $U_{g,k+1}=U_{g,k}$. It is evident, from the
data reported in table 2, that we have achieved six decimal places of
agreement.

As mentioned before, the case $P=1$ is the Blasius problem (1). In this case
our non-ITM reduces to the original method defined by Töpfer [40]. In that
case, the computed skin friction coefficient value, namely $0.332057336215$,
obtained with $\Delta\eta=0.001$ and $\eta^{*}_{\infty}=10$, is in good
agreement with the values available in literature, see for instance the value
$0.332057336215$ computed by Fazio [7] by a free boundary formulation of the
Blasius problem or the value $0.33205733621519630$ computed by Boyd [3] who
believes all the decimal digits to be correct.

##  4 Concluding remarks.

The main contribution of this paper is the extension of the non-ITM, proposed
by Töpfer [40] and defined for the numerical solution of the celebrated
Blasius problem [2], to an extended Blasius problem. This method allows us to
solve numerically the extended Blasius problem by solving a related initial
value problem and then rescaling the obtained numerical solution. The obtained
numerical results, have been improved both by a mesh refinement and using the
Richardson’s extrapolation technique. In this way, we can be confident that
the computed six decimal place are correct.

Acknowledgement. The research of this work was partially supported by the
University of Messina and by the GNCS of INDAM.

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# High-temperature superconductivity in A15-type La4H23 below
$100\text{\,}\mathrm{GPa}$

Sam Cross<EMAIL_ADDRESS>HH Wills Laboratory, University of Bristol,
Bristol, BS8 1TL, UK  Jonathan Buhot  HH Wills Laboratory, University of
Bristol, Bristol, BS8 1TL, UK  Annabelle Brooks  HH Wills Laboratory,
University of Bristol, Bristol, BS8 1TL, UK  William Thomas  HH Wills
Laboratory, University of Bristol, Bristol, BS8 1TL, UK  Annette Kleppe
Diamond Light Source, Chilton, Oxfordshire, OX11 0DE, UK  Oliver Lord
Department of Earth Sciences, University of Bristol, Bristol BS8 1RJ, UK  Sven
Friedemann<EMAIL_ADDRESS>HH Wills Laboratory, University of
Bristol, Bristol, BS8 1TL, UK

###### Abstract

High-temperature superconductivity has been observed in binary hydrides such
as LaH10 at pressures above $\sim$150\text{\,}\mathrm{GPa}$$. Hydrogen cage
structures have been identified as a common motif beneficial for high critical
temperatures $T_{\text{c}}$. Efforts are now focused on finding hydride high-
temperature superconductors at lower pressures. We present evidence for high-
temperature superconductivity in binary La4H23 with A15-type structure
featuring hydrogen cages at a pressure of $P=$95\text{\,}\mathrm{GPa}$$. We
synthesise La4H23 from a lanthanum film capped with a palladium catalyst
promoting the dissociation of hydrogen. In resistance measurements, we observe
superconductivity with a transition temperature
$T_{\text{c}}\sim$90\text{\,}\mathrm{K}$$. In X-ray diffraction on the same
sample, we identify the A15-type cubic body centred structure of the lanthanum
sublattice. From comparison with earlier XRD and structure prediction studies,
we identify this phase with La4H23. Our study reinforces the concept of
hydrogen cages for high-temperature superconductivity.

##  I Introduction

High-temperature superconductivity up to $\sim$250\text{\,}\mathrm{K}$$ has
been reported and independently confirmed in LaH10 at pressures $\sim$
$180\text{\,}\mathrm{GPa}$ [1, 2]. In LaH10 and other hydride high-temperature
superconductors, clathrate-like cages of hydrogen atoms have been identified
as a common structural motif [3, 4]. Recent efforts have been devoted to
finding high-temperature superconducting phases at lower pressures [4].
Notably, the cubic $Fm\bar{3}m$ phase of the LaH10 system could be stabilised
at moderate pressures around $110\text{\,}\mathrm{GPa}$ in compounds
synthesised from an alloy of cerium and lanthanum [5]. Recent XRD studies
found new structures in the binary lanthanum hydride system that feature
hydrogen cages and hence are strong candidates for superconductivity [6].

The A15-type structure (space group $Pm\bar{3}n$) is expected to be favourable
for superconductivity as it promotes clathrate cages of hydrogen atoms around
the body-centred cubic lattice of the metal atoms as illustrated in the inset
of Figure 1 (c). In this structure, hydrogen atoms feature bond lengths in the
range
$\approx$1.1\text{\,}\mathrm{\SIUnitSymbolAngstrom}1.3\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$
favourable for high-temperature superconductivity. This bond length is much
larger than in hydrogen molecules at this pressure and indicates a weaker bond
compared to molecular bonding. Through the donation of electrons from the
metal atoms, the hydrogen anitbonding bands can be populated leading to
stabilisation of the hydrogen cages and providing an avenue for metallisation
and superconductivity [7]. The weakening of the H-H bond and the formation of
cages is required for hydrogen to contribute electronic states at the Fermi
energy and hence for significant electron-phonon coupling involving the
hydrogen atoms required for high-temperature superconductivity.

Hydride compounds of the A15-type structure have been searched for
superconductivity in computational studies, e.g., leading to the prediction of
high-temperature superconductivity in LiNi3H23 [8]. Recently,
superconductivity was indeed discovered in members of this structure type, in
Lu4H23 at pressures above $200\text{\,}\mathrm{GPa}$ [9]. Furthermore, the
structure type has also been detected in X-ray diffraction studies of Ba4H23,
Eu4H23, and La4H23 [10, 11, 6]. Hence, there is a strong interest to probe
experimentally for superconductivity in these members of the A15 structure
type.

The A15-type structure has recently been detected in the lanthanum hydride
binary system with single-crystal XRD by Laniel et al. at pressures between
$96\text{\,}\mathrm{GPa}150\text{\,}\mathrm{GPa}$ [6]. Laniel et al.
synthesised La4H23 by laser heating bulk lanthanum with paraffin oil as a
hydrogen donor material. During the preparation of this manuscript, Guo et al.
reported synthesis of La4H23 from bulk LaH3 and ammonia borane (NH3BH3).
Furthermore, Guo et al. detected superconductivity in their samples of La4H23
in resistance measurements. Here, we present synthesis of La4H23 from a thin
film of lanthanum capped with palladium and using NH3BH3 as a hydrogen donor
[12, 13]. Upon laser heating, molecular hydrogen is dissociated by the
palladium and diffuses into the lanthanum forming La4H23. We detect the
formation of La4H23 with XRD measurements and superconductivity through four-
point resistance measurements including the characteristic suppression in
magnetic fields up to $12\text{\,}\mathrm{T}$.

##  II Synthesis of La4H23

The successful synthesis of La4H23 is evident from our X-ray diffraction in
Figure 1. We observe characteristic reflections between
$$7.5\text{\,}\mathrm{\SIUnitSymbolDegree}$\leq
2\theta\leq$10\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and
$$13.5\text{\,}\mathrm{\SIUnitSymbolDegree}$\leq
2\theta\leq$16\text{\,}\mathrm{\SIUnitSymbolDegree}$$ that are best assigned
to the La4H23 phase. Besides the La4H23, we also observe peaks from elemental
lanthanum (orthorombically distorted $Fmmm$) and gold ($Fm\bar{3}m$). The
former is residue from the synthesis of La4H23 whilst the latter originates
from the electrodes used for our resistance measurements. We can rule out
several other phases including $I4/mmm$ LaH4 (Figure E1), $P6_{3}/mmc$ LaH9
(Figure E2) and $C2/m$ LaHx (Figure E3) that have been reported for the La-H
binary system as detailed in the extended data. We can also rule out the
formation of $Fm\bar{3}m$ LaBH8 (Figure E4) as well as the carbohydride
impurity $P6_{3}/mmc$ LaCH2 reported by Laniel et al. at
$96\text{\,}\mathrm{GPa}$ (Figure E5) [6].

Figure 1: Detection of the A-15 structure in X-Ray diffraction. (a) Rietveld
and Le Bail refinements of La4H23 to the observed XRD pattern shown by the
solid red and grey lines respectively. Data between
$11\text{\,}\mathrm{\SIUnitSymbolDegree}13\text{\,}\mathrm{\SIUnitSymbolDegree}$
are dominated by residual elemental lanthanum in the orthorombically distorted
$Fmmm$ phase and gold in the $Fm\bar{3}m$ phase originating from the
electrodes. No La4H23 reflections are expected in this range. (b) Azimuthal
(cake) plot representation of the area detector image in grayscale, where high
and low intensity are represented in white and black respectively. Sample
reflections appear as white vertical lines. (c) Unit cell volume as a function
of pressure for the La4H23 phase including data from this work, and that of
references [6, 14]. The A15 clathrate structure is shown in the inset,
generated using the software Vesta [15].

The XRD data reveal a homogeneous sample without texture. This is visible from
both the azimuthal representation of the detector image in Figure 1(b) as well
as the analysis of the angle-integrated XRD pattern. The detector image shows
continuous vertical lines without significant spots indicating a large
ensemble of grains in a polycrystalline sample. At the same time, the angle-
integrated XRD pattern exhibits peak intensities as expected for a powder
average (Figure 1 (a)). Indeed, we are able to fit the observed XRD pattern
with a Rietveld refinement without any preferential orientation, only invoking
strain to reproduce the peak broadening.

The lattice constants extracted from Rietveld refinement provide confidence in
the assignment of the La4H23 phase with A15-type La sublattice and suggest a
large hydrogen content close to the stoichiometric ratio La:H = 1:5.75 for the
La4H23 phase. The lattice parameters
($a=$6.191(4)\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$, volume per La atom =
$29.66(8)\text{\,}{\mathrm{\SIUnitSymbolAngstrom}}^{3}$) are in good agreement
with previous reports and close to the predicted equation of state (EoS) (see
Figure 1(c)) [6]. The comparison with the EoS suggests a stoichiometric La4H23
composition. However, some uncertainty remains for this indirect method of
determining the hydrogen content as pointed out by Laniel et al. [6]. In
particular, we cannot exclude vacancies on the hydrogen lattice and have
limited resolution to detect the subtle modifications of the metal sublattice
that might be associated with hydrogen vacancies and superstructures [10].

##  III High-Temperature Superconductivity in La4H23

The resistance measurements presented in Figure 2 provide evidence for high-
temperature superconductivity in La4H23. Immediately after laser heating, we
observe a pronounced drop in the resistance with an onset at
$T_{\text{c}}=$86\text{\,}\mathrm{K}$$ and reaching zero resistance within the
resolution $|R-R_{\text{n}}|/R_{\text{n}}\leq$1\text{\times}{10}^{-3}$$ of our
experiment below $60\text{\,}\mathrm{K}$, where $R_{\text{n}}$ is the
resistance of the normal state. We have repeated resistance measurements
several times including before and after XRD measurements and find the
superconductivity to be stable with a slight increase to
$T_{\text{c}}=$95\text{\,}\mathrm{K}$$ between subsequent measurements. We
attribute this increase to annealing of the sample at room temperature similar
to reports on H3S [16].

Figure 2: Detection of superconductivity in La4H23and resistance measurements
on subsequent cooling runs. In each case, cooling and warming measurements are
indicated. The value of $T_{\text{c}}$ was determined at the onset of
superconductivity from the intersection of linear fits to the normal and
superconducting states, shown by the dashed line in the third thermal cycle.
Cooldown and warmup 1 and 2 were collected shortly after laser heating.
Cooldown and warmup 3 were recorded after the XRD measurements.

The normal state resistance shows indications of phase instabilities with
decomposition exhibiting activated behaviour. If warming the sample above
$250\text{\,}\mathrm{K}$, the sample resistance increases over time as
illustrated in Figure 2. This manifests as an initial increase during
cooldowns and further increase during warm-up. Only below
$250\text{\,}\mathrm{K}$, cool-down and warm-up agree where we observe
metallic behaviour in the sample immediately after laser heating (thermal
cycles 1 and 2).

The changes to the sample also induce fundamental change from metallic to non-
metallic over time. Over multiple warm-ups to above $250\text{\,}\mathrm{K}$,
the resistance changes from metallic ($\text{d}R/\text{d}T>0$) to non-metallic
($\text{d}R/\text{d}T<0$). A similar change from metallic to non-metallic has
been observed by Guo et al. after the first laser heating and in subsequent
decompression runs. In our study, the pressure before and after laser heating
remained stable within our uncertainty of $\pm$ $5\text{\,}\mathrm{GPa}$,
therefore the change in the normal state could originate from either small
changes in pressure and/or could be time dependent. Equally, the broadening of
the superconducting transition and the loss of zero resistance reported by Guo
et al. below $100\text{\,}\mathrm{GPa}$ may be related to sample
decomposition.

The combination of increasing resistance and change to non-metallic behaviour
suggests that part of the sample slowly transforms to a semimetallic or
insulating material. In our sample, we have detected La4H23 and elemental La
with XRD, both of which may form part of the electrical path sampled by our
resistance measurements with La4H23 providing at least a percolative path
supporting zero resistance of the high-$T_{\text{c}}$ state. Hence, the
increase in resistance with time above $250\text{\,}\mathrm{K}$ could be a
conversion of either La4H23 or La to a semimetallic or insulating phase, e.g.
LaH3 which has been reported to become semimetallic at high pressures [2]. We
argue that La4H23 is converting because Guo et al. observe a very similar
change from metallic to non-metallic but don’t have elemental lanthanum in
their pressure cell. We note that LaH3 has been ruled out as a
high-$T_{\text{c}}$ material [2] and could remain undetected in XRD in the
presence of La4H23 as shown in Figure E6. In our sample the volume fraction of
La4H23 whilst decreasing, retains a full path to detect zero resistance during
the course of our measurements whilst in the study of Guo et al., the non-zero
resistance below $100\text{\,}\mathrm{GPa}$ suggests that a larger portion of
the sample has transformed to a semimetallic state.

The change in resistance may be related to the thermodynamic and/or dynamical
instabilities found in DFT calculations by Guo et al. [14]. At
$100\text{\,}\mathrm{GPa}$, the DFT predicts that La4H23 is above the convex
hull by a small energy. Such an energy scale could lead to a disintegration
with an activation energy. In addition, DFT predicts imaginary phonons that
indicate that the phase is inherently unstable. Such imaginary phonons are in
contradiction with the observed presence of La4H23 in the studies by Laniel,
Guo, and in the present work. Hence, a mechanism like a superstructure of the
hydrogen lattice must exist that stabilises La4H23 against dynamical
instabilities. This mechanism might still keep the material metastable with a
low energy barrier comparable to the activation energy of
$\sim$250\text{\,}\mathrm{K}$$ observed here.

Further investigations are required to map the stability range of the La4H23
phase including under the influence of excess or deficiency of hydrogen in the
DAC. So far, all reported syntheses have employed hydrogen donor materials
like NH3BH3 and paraffin. It will be of interest to check if a synthesis in
excess molecular hydrogen shows a larger stability range to lower pressures.
Further X-ray diffraction that check for superstructures and decomposition are
equally desirable but challenging.

The suppression of the resistive transition in magnetic field provides further
evidence for superconductivity in La4H23. The transition is suppressed by
$13\text{\,}\mathrm{K}$ in a magnetic field of $12\text{\,}\mathrm{T}$ as
illustrated in Figure 3 (a). We extract $T_{\text{c}}(H)$ in order to map the
temperature dependence of the upper critical field $H_{\text{c}2}(T)$ as shown
in Figure 3(b), where the zero-temperature value $H_{\text{c}2}(0)$ is
estimated from extrapolations of the Ginzburg-Landau (GL) and WHH models,
yielding $47\text{\,}\mathrm{T}$ and $63\text{\,}\mathrm{T}$ respectively. The
linear regime of the critical field $H_{\text{c}2}(T)$ can be used to estimate
the zero-temperature coherence length $\xi_{0}$ of the superconducting state
and the Fermi velocity $v_{\text{F}}$ of the normal state. Using the clean-
limit expression, and the slope of the critical field near $T_{\text{c}}$
($\text{d}H_{\text{c}2}/\text{d}T$ = -0.87 TK-1) we obtain
$\xi_{0}=$2.7\text{\,}\mathrm{nm}$$. The change of the resistance to non-
metallic indicates that the sample requires a dirty-limit expression. In the
absence of direct or indirect measurements that allow to estimate the mean
free path, the dirty-limit expressions cannot be used to extract $\xi_{0}$.
However, the clean-limit expression still provides a good estimate and likely
only small corrections will arise from dirty-limit behaviour. Indeed the
comparable slope of $H_{\text{c}2}(T)$ observed by Guo et al. suggests that
both studies observe the intrinsic behaviour of La4H23.

The Fermi velocity is estimated as
$v_{\text{F}}=$1.8\text{\times}{10}^{5}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$.
This value is lower compared to H3S [17]. Likely, this reflects a reduced
renormalisation and hence a weaker electron-phonon coupling compared to H3S in
agreement with the lower $T_{\text{c}}$. A more accurate estimation of the
renormalisation requires detailed band structure calculations based on a
stable structure.

Figure 3: Superconducting critical field of La4H23. (a) Temperature sweeps for
different magnetic fields. The value of $T_{\text{c}}$ was determined at the
onset of superconductivity as shown by the dashed line for the
$12\text{\,}\mathrm{T}$ curve. (b) Temperature dependence of the upper
critical field with Ginzburg-Landau and WHH fits to estimate H${}_{c2}(0)$,
and a linear fit to extract the gradient near $T_{\text{c}}$.

##  IV Conclusion

The discovery of high-temperature superconductivity in La4H23 with the A15
structure reinforces that hydrogen cage structures are beneficial for
superconductivity. The A15-type structure is so far the prime example of how
high-temperature superconductivity can be realised below
$100\text{\,}\mathrm{GPa}$ in a binary hydride material.

The XRD analysis suggests that La4H23 is the major phase synthesised at
pressures around $100\text{\,}\mathrm{GPa}$. This may help forming pure
samples in the future to study the intrinsic superconducting properties and
hence has the potential to advance the understanding of high-temperature
superconductivity in hydride compounds.

##  V Methods

High pressures were achieved in a DAC machined from MP35N alloy, with diamond
anvils of culet diameter $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ bevelled
at $8\text{\,}\mathrm{\SIUnitSymbolDegree}$ to a diameter of
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Gaskets were prepared by pre-
indentation of $250\text{\,}\mathrm{\SIUnitSymbolMicro m}$ steel 301 and
insulated with a boron nitride and epoxy mixture. The insulation was indented
to a thickness $\sim$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and a sample
space of diameter $28\text{\,}\mathrm{\SIUnitSymbolMicro m}$ was drilled. The
anvil with the sample was first coated in $50\text{\,}\mathrm{nm}$ of alumina
to provide a thermal insulation during laser heating and to help prevent
diamond embrittlement from hydrogen diffusion. For electrical measurements,
five bi-layer tungsten-gold electrodes were sputtered/evaporated directly onto
one of the diamond anvils.

A lanthanum film of thickness $260\text{\,}\mathrm{nm}$ was deposited onto the
electrodes in a vacuum $\sim$1\text{\times}{10}^{-7}\text{\,}\mathrm{mbar}$$
after multiple pre-melts to remove oxide impurities from the La source. The La
film was capped with a $5\text{\,}\mathrm{nm}$ layer of palladium to act as an
oxidation barrier, and to catalyse hydrogen diffusion into the La sample [18].

Due to the hygroscopic nature of AB, we first purified by sublimation to
remove any residual moisture and impurities that would react with the La film.
Loading of the AB into the gasket indentation and initial cell closing were
done in an inert Ar atmosphere in a glove box with residual O2/H2O $<$ 0.5
ppm. The sample was pressurised to 95 GPa with NH3BH3 serving as the pressure
transmitting medium (PTM).

The pressure was determined from the Raman shift of the diamond edge using the
Akahama scale as shown in Figure E7 [19]. The unit cell volumes extracted from
Le Bail refinement of residual lanthanum (Fmmm) and gold (Fm$\overline{3}$m)
(Figure E8) from the electrodes are consistent with the pressure measured
using the diamond edge [20, 21].

The La film was characterised in resistance measurements prior to laser
heating (Figure E9), confirming metallic behaviour with residual resistance
ratio ($RRR$) $\sim$ 2.5. The superconducting transition of pure La is
observed with $T_{\text{c}}$ = $8.6\text{\,}\mathrm{K}$ at
$95\text{\,}\mathrm{GPa}$ which is $\sim$1\text{\,}\mathrm{K}$$ higher than
reported in previous studies [20], and may originate from a pressure gradient
across the sample (cf. sample pictures in Figure E10).

Synthesis of La4H23 was achieved through heating of the sample and hydrogen
donor material AB. The hydrogen released from AB is dissociated at the
palladium capping layer and diffuses into the lanthanum film where it reacts
to form La4H23. Heating of the sample was done using a
$1064\text{\,}\mathrm{nm}$ YAG laser with multiple $300\text{\,}\mathrm{ms}$
pulses. Laser power was incremented after each successive pulse until glowing
was observed indicating a temperature of $\sim$1000\text{\,}\mathrm{K}$$. The
four-point resistance was monitored during heating, where the normal state
resistance increased by a factor $\sim$ 20 indicating chemical reaction
between the La film and the AB. No further heating of the sample was carried
out after the first observed glowing in order to preserve the electrodes,
which likely explains the residual lanthanum observed in XRD.

X-ray diffraction was carried out at the I15 extreme conditions beamline at
Diamond Light Source with a Pilatus3 X CdTe 2M area detector, wavelength
$\lambda=$0.425\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ (sample-to-detector
distance $385\text{\,}\mathrm{mm}$). The beam was collimated to
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The calibrant used was LaB6.
Integration of the powder pattern and removal of spurious reflections were
done in Dioptas [22]. Background subtraction and structural refinements were
done in GSAS II [23].

Electrical transport measurements were done in a four-point configuration with
an excitation current of $10\text{\,}\mathrm{\SIUnitSymbolMicro A}$ using
either an SRS SIM921 AC resistance bridge or an SR830 lock-in amplifier.
Magnetic field studies were done at the University of Bristol with an Oxford
Instruments $14\text{\,}\mathrm{T}$ magnet.

###### Acknowledgements.

The authors are grateful for discussions with Lewis Conway. This work was
carried out with the support of Diamond Light Source, instrument I15 (proposal
CY35426-1). This work was supported by ERC Horizon 2020 programme under grant
715262-HPSuper and the EPSRC under grants EP/V048759/1.

## Additional information

Data will be made available at the University of Bristol data repository

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## Extended Data

Figure E1: Peak positions and relative intensities for $I4/mmm$ LaH4,
generated with lattice parameters reported by Chen et al. at
$108\text{\,}\mathrm{GPa}$ [5]:
$a=b=$2.918\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ and
$c=$5.841\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$. To our knowledge, there
has been no experimental observation of the $I4/mmm$ phase below this
pressure. Figure E2: Peak positions and relative intensities for $P6_{3}/mmc$
LaH9-10, generated with lattice parameters reported by Chen et al. at
$89\text{\,}\mathrm{GPa}$ [5]:
$a=b=$3.748\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ and
$c=$5.833\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$. Figure E3: Peak positions
and relative intensities for $C2/m$ LaHx, generated with lattice parameters
reported by Chen et al. at $89\text{\,}\mathrm{GPa}$ [5]:
$a=$5.310\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$,
$b=$5.748\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$,
$c=$3.893\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ and
$\beta=$92.2\text{\,}\mathrm{\SIUnitSymbolDegree}$$. Figure E4: Peak positions
and relative intensities for $Fm\bar{3}m$ LaBH8, generated with lattice
parameters calculated by Belli and Errea at $100\text{\,}\mathrm{GPa}$ [24]:
$a=$5.311\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$. Figure E5: Peak positions
and relative intensities for $P6_{3}/mmc$ LaCH2 as reported by Laniel et al.
at $96\text{\,}\mathrm{GPa}$:
$a=b=$3.4936\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$,
$c=$4.9430\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$. Figure E6: Peak positions
and relative intensities for $Fm\bar{3}m$ LaH3, generated with lattice
parameter $a=$4.82\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ estimated from the
equation of state given in reference [6]. Figure E7: The determination of the
pressure before laser heating from the Raman spectrum of the diamond edge
using the Akahama scale [19]. Figure E8: Le Bail and Rietveld refinement of
the XRD pattern. The volume per lanthanum atom of the pure La ($Fmmm$) is
$16.99(9)\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$ in agreement with
previous calculated and experimental reports, shown in the inset [6, 20].
Figure E9: Resistance versus temperature of the La + Pd film at
$95\text{\,}\mathrm{GPa}$ before laser heating, displaying metallic behaviour
with $RRR\sim$ 2.5 and superconductivity with onset $T_{\text{c}}$=
$8.6\text{\,}\mathrm{K}$. Figure E10: Sample photographs before laser heating
in (a) transmission and (b) reflection. The sample after laser heating is
shown in (c) and (d).