# On the local negativity of Enriques and K3 surfaces

Roberto Laface and Piotr Pokora  Roberto Laface
Institut für Algebraische Geometrie, Leibniz Universität Hannover,
Welfengarten 1, D-30167 Hannover, Germany<EMAIL_ADDRESS>Piotr
Pokora
Instytut Matematyki, Pedagogical University of Cracow, Podchora̧żych 2,
PL-30-084 Kraków, Poland. Institut für Mathematik, Johannes Gutenberg
Universität Mainz, Staudingerweg 9, D-55099 Mainz, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this note we study the local negativity for certain configurations of
smooth rational curves on smooth K3 and Enriques surfaces. We show that for
such rational curves there is a bound for the so-called local Harbourne
constants, which measure the local negativity phenomenon. Moreover, we provide
explicit examples of interesting configurations of rational curves on some K3
and Enriques surfaces and compute their local Harbourne constants.

###### Key words and phrases:

rational curve configurations, algebraic surfaces, Miyaoka inequality, blow-
ups, negative curves, bounded negativity conjecture, Harbourne constants

###### 2010 Mathematics Subject Classification:

Primary 14C20; Secondary 14J70

##  1\. Introduction

In this note we continue studies on the local negativity of algebraic
surfaces. In last years there is a resurgence around questions related to
negative curves on algebraic surfaces. One of the most challenging problems is
the Bounded Negativity Conjecture (BNC in short).

######  Conjecture 1.1 (Bounded Negativity Conjecture).

Let $X$ be a smooth projective surface defined over a field of characteristic
zero. Then there exists an integer $b(X)\in\mathbb{Z}$ such that for all
_reduced_ curves $C\subset X$ one has $C^{2}\geqslant-b(X)$.

It is easy to see that the number $b(X)$ depending on $X$ can be arbitrary
large. In order to see this phenomenon consider the blow up $X_{s}$ of the
projective plane $\mathbb{P}^{2}_{\mathbb{C}}$ along $s>>0$ mutually distinct
points $P_{1},\dots,P_{s}$ lying on a line $l$. It is easy to see that the
strict transform of $l$ has the form $\widetilde{l}_{s}:=H-E_{1}-\cdots-
E_{s}$, where $H$ is the pull back of
$\mathcal{O}_{\mathbb{P}^{2}_{\mathbb{C}}}(1)$ and $E_{1},\dots,E_{s}$ are the
exceptional divisor, and $\widetilde{l}_{s}^{2}=-s+1$. Moreover, it is not
difficult to see that $b(X_{s})=-s+1$. In order to avoid such situations we
can define an asymptotic version of self-intersection numbers, in the case of
arbitrary blow ups of the projective plane one can divide the self-
intersection number of a reduced curve by the number of point we blown up our
surface. It turns out that this approach is more effective in the context of
the BNC.

It is worth pointing out that the BNC is widely open, but there are some cases
for which we know $b(X)$. It can be shown that the BNC is true for minimal
models with Kodaira dimension equal to zero. In particular, we know that for
K3 or Enriques surfaces every reduced and irreducible curve $C$ has
$C^{2}\geqslant-2$ by the adjuction formula. However, it is not know whether
the BNC still holds for blow-ups of those surfaces along sets of points.

The main aim of this note is to study the BNC for blow ups of K3 and Enriques
surfaces from the point of view of Harbourne constants, which were introduced
in [2], and allow to measure the negativity phenomenon asymptotically. Before
we define the main object of this paper, we need to recall some standard
notions.

######  Definition 1.2 (The numbers $t_{i}$).

Let $\mathcal{C}$ be a configuration of finitely many mutually distinct smooth
curves on a projective surface $X$. We say that $P$ is an $r$-fold point of
the configuration $\mathcal{C}$, if it is contained in exactly $r$ irreducible
components of $\mathcal{C}$. The union of all $r$-fold points
$P\in\mathcal{C}$ for $r\geqslant 2$, is the singular set $\mathop{\rm
Sing}\nolimits(\mathcal{C})$ of $\mathcal{C}$. We set the number
$t_{r}=t_{r}(\mathcal{C})$ to be the number of $r$-fold points in
$\mathcal{C}$.

We will mostly deal with configurations of smooth curves having only
transversal intersection points. Letting $\mathcal{C}=\\{C_{1},\dots,C_{n}\\}$
be a arrangement of such curves on $X$, consider the blow up of $X$ along the
set $\mathop{\rm Sing}\nolimits(\mathcal{C})$, with the exceptional divisors
$E_{1},\ldots,E_{s}$ ($s=\\#{\rm Sing}(\mathcal{C})$). Let
$C:=C_{1}+\cdots+C_{n}$, and let $\widetilde{C}$ be the strict transform of
$C$. Then the divisor

$\widetilde{C}+E_{1}+\cdots+E_{s}$

is a simple normal crossing divisor on $Y$, meaning that

  1. (1)

it is reduced;

  2. (2)

its irreducible components are all smooth;

  3. (3)

there are at most two irreducible components going through a point of the
divisor (i.e. at the singular points, the divisor locally looks like the
intersection of the coordinate axes in $\mathbb{C}^{2}$).

Simple normal crossing divisors come pretty handy, as it is quite easy to
compute their Chern numbers, a fact that we will employ in our computations
(see the proof of Theorem 2.1). In the present note, we are interested in
Harbourne constants attached to transversal arrangements, i.e. arrangements of
curves such that all singular points are intersection points of irreducible
components and these singular points are transversal. They can be viewed as a
way to measure the average negativity coming from singular points in the
arrangement.

######  Definition 1.3 (Harbourne constants of a transversal arrangement).

Let $X$ be a smooth projective surface. Let $D=\sum_{i=1}^{\tau}C_{i}$ be a
transversal arrangement of curves on $X$ with $s=s(D)$. The rational number

$h(X;D)=h(D)=\frac{1}{s}\left(D^{2}-\sum\limits_{P\in\mathop{\rm
Sing}\nolimits(D)}m_{P}^{2}\right),$ (1)

where $m_{P}$ is the multiplicity of the divisor $D$ at the point $P$, is the
_Harbourne constant of the transversal arrangement_ $D\subset X$.

We can also express the Harbourne constant in terms of the $t_{i}$’s as
follows:

$h(D)=\frac{1}{s}\left(D^{2}-\sum\limits_{r\geqslant 2}r^{2}t_{r}\right).$

##  2\. A bound on local Harbourne constants on Enriques and K3 surfaces

We would like to focus on the case of configurations of smooth rational curves
on K3 and Enriques surfaces having only transversal intersection points. We
start with the following result which is a generalization of a result due to
Miyaoka [12, Section 2.4].

######  Theorem 2.1.

Let $X$ be a smooth complex projective K3 or Enriques surface and let
$\mathcal{C}\subset X$ be a configuration of smooth rational curves having $n$
irreducible components and only transversal intersection points. Then

$4n-t_{2}+\sum_{r\geqslant 3}(r-4)t_{r}\leqslant 72.$

###### Proof.

Let $\mathcal{C}=\\{C_{1},\dots,C_{n}\\}$ be a configuration of smooth
rational curves on $X$. If $\text{Sing}(\mathcal{C})$ denotes the set of
singular points of the configuration, we define $S=\\{p_{j}\\}_{j=1}^{k}$ to
be the subset of points in $\text{Sing}(\mathcal{C})$ with multiplicity
$\geqslant 3$. Consider the blow-up of $X$ at the points of $S$, namely

$\sigma:Y\longrightarrow X;$

under pull-back along $\sigma$, the configuration $\mathcal{C}$ on $X$ yields
a configuration $\sigma^{*}\mathcal{C}$ which consists of the strict
transforms of the $C_{i}$’s and the exceptional divisors. Notice that
$\sigma^{*}\mathcal{C}$ is again a configuration of smooth rational curves
that admits double points only as singularities. Following [12, Section 2.4],
we set $L:=C_{1}+\cdots+C_{n}$ and
$M:=\widetilde{C}_{1}+\cdots+\widetilde{C}_{n}$. The idea is to use the
Miyaoka-Yau inequality

$3c_{2}(Y)-3e(M)\geqslant(K_{Y}+M)^{2},$

and thus we now need to compute the terms in the above inequality. We see that

$\displaystyle c_{2}(Y)=c_{2}(X)+k,$ $\displaystyle e(M)=2n-t_{2},$

which yield $c_{2}(Y)-e(M)=c_{2}(X)+k-2n+t_{2}$. The assumption on $X$ being
either a K3 surface or an Enriques surface implies in particular that the
canonical divisor $K_{X}$ is numerically trivial. Therefore, if
$E:=\sum_{j=1}^{k}E_{j}$ is the sum of all exceptional divisors, we have

$\displaystyle K_{Y}+M=\sigma^{*}(K_{X}+L)-\sum_{j=1}^{k}(m_{j}-1)E_{j},$
$\displaystyle(K_{Y}+M)^{2}=L^{2}-\sum_{j=1}^{k}(m_{j}-1)^{2}.$

Notice that

$K_{Y}+M=(\sigma^{*}K_{X}+E)+M=\sigma^{*}K_{X}+(E+M),$

and as $K_{X}$ is numerically trivial, also $\sigma^{*}K_{X}$ is, and thus
$K_{Y}+M$ is numerically equivalent to an effective divisor. This allows us to
us the Miyaoka-Yau inequality according to [12, Cor. 2.1]. Also,

$\displaystyle L^{2}$ $\displaystyle=-2n+2\sum_{i<j}C_{i}.C_{j}$
$\displaystyle=-2n+2\sum_{r\geqslant 2}{r\choose 2}t_{r}$
$\displaystyle=-2n+2t_{2}+2\sum_{r\geqslant 3}{r\choose 2}t_{r}.$

It follows that

$\displaystyle(K_{Y}+M)^{2}$ $\displaystyle=-2n+2t_{2}+2\sum_{r\geqslant
3}{r\choose 2}t_{r}-\sum_{j=1}^{k}(m_{j}-1)^{2}$
$\displaystyle=-2n+2t_{2}+2\sum_{j=1}^{k}{m_{j}\choose
2}-\sum_{j=1}^{k}(m_{j}-1)^{2}$
$\displaystyle=-2n+2t_{2}+\sum_{j=1}^{k}(m_{j}-1).$

By plugging in the Miyaoka-Yau inequality, we see that

$\displaystyle 3c_{2}(X)$ $\displaystyle\geqslant
4n-t_{2}-3k+\sum_{j=1}^{k}(m_{j}-1)$
$\displaystyle=4n-t_{2}+\sum_{j=1}^{k}(m_{j}-4)$
$\displaystyle=4n-t_{2}+\sum_{r\geqslant 3}(r-4)t_{r},$

and the result follows from the fact $c_{2}(X)=24$ for a K3 surface and
$c_{2}(X)=12$ for an Enriques surface. ∎

Now we can prove the main result of this paper.

######  Theorem 2.2.

Let $X$ be a smooth complex projective K3 or Enriques surface and let
$\mathcal{C}$ be a configuration of smooth rational curves having $n$
irreducible components and only $s\geqslant 1$ transversal intersection
points. Then

$h(X;\mathcal{C})\geqslant-4+\frac{2n+t_{2}-72}{s}$

###### Proof.

If $\mathcal{C}=\\{C_{1},\dots,C_{n}\\}$, the Harbourne constant is computed
by $\widetilde{L}^{2}/s$, where $L:=C_{1}+\cdots+C_{n}$, and $\widetilde{L}$
is its strict transform in the blow-up at the $s$ singular points of the
configuration. We observe that

$\widetilde{L}^{2}/s=\frac{L^{2}-\sum_{j}m_{j}^{2}}{s}=\frac{-2n+I_{d}-\sum_{r\geqslant
2}r^{2}t_{r}}{s},$

where $I_{d}:=2\sum_{i<j}C_{i}.C_{j}$ is the number of incidences of the
collection $\mathcal{C}$ of rational curves on $X$. The proof of [14, Main
Theorem] shows that

$I_{d}-\sum_{r\geqslant 2}r^{2}t_{r}=-\sum_{r\geqslant 2}rt_{r},$

and moreover, by arguing in a similar way, we can rephrase the bound in
Theorem 2.1 in the following way:

$-\sum_{r\geqslant 2}rt_{r}\geqslant-4s+4n+t_{2}-72.$

This yields

$h(X;\mathcal{C})\geqslant-4+\frac{2n+t_{2}-72}{s},$

and we are done. ∎

######  Remark 2.3.

For configurations of rational curves on Enriques surfaces, Theorems 2.1 and
2.2 yield even stronger bounds: as the topological Euler characteristic of any
Enriques surface is $12$, we obtain sharper bounds by replacing every
occurrence of $72$ by $36$ (recall that for a K3 surface $X$ we have
$3c_{2}(X)=72$).

Let us define the following number.

######  Definition 2.4.

Let $X$ be a smooth complex projective K3 or Enriques surface. The real number

$H_{\text{rational}}(X)=\inf_{D}h(X;D),$

where the infimum is taken over all transversal arrangements of smooth
rational curves $D\subset X$ is the _global rational Harbourne constant_ of
$X$.

######  Corollary 2.5.

Let $X$ be a smooth complex projective K3 or Enriques surface. Then

$H_{\text{rational}}(X)\geqslant-72.$

##  3\. Examples

We now give a few examples of interesting configurations of smooth rational
curves on K3 and Enriques surfaces. We will use Theorem 2.1 to give a lower
bound for the Harbourne constants.

######  Example 3.1 (Six general lines in $\mathbb{P}^{2}$).

In the complex projective plane $\mathbb{P}^{2}$, consider six lines in
general position, and denote this configuration by $\mathcal{L}$.

Figure 1. Six lines in general position in $\mathbb{P}^{2}$.

This configuration has only double points as singularities, and their number
is the maximum possible of 15. Let $Y$ denote the 2:1 cover of
$\mathbb{P}^{2}$ branched over the configuration $\mathcal{L}$: it is a normal
surface with 15 singularities of type $\rm A_{1}$, namely the points sitting
over the intersection of the six lines in $\mathcal{L}$. We can resolve the
singularities of $Y$ by blowing-up once at each singular point; this yields a
smooth surface $X$, which is a K3 surface by general theory. Alternatively, we
could have first blown-up $\mathbb{P}^{2}$ at the singular points of
$\mathcal{L}$, and then taken a 2:1 cover branched over the strict transforms
of the six lines (which are disjoint after performing a blow-up).

On the K3 surface $X$, we have a new configuration of curves, which we call
$\mathcal{C}$, given by the union of the strict transforms of the six lines
and the exceptional divisors. Notice that $\mathcal{C}$ consists of $6+15=21$
($-2$)-curves which intersect at 30 points of multiplicity two, thus $n=21$,
$t_{2}=30$ and $t_{r}=0$ for all $r\geqslant 3$. We can compute the Harbourne
constant:

$h(X;\mathcal{C})=\frac{18-120}{30}\approx-3.4666,$

which together with the lower bound of Theorem 2.2 yields

$-4\leqslant h(X;\mathcal{C})=-102/30\approx-3.4666.$

######  Example 3.2 (Vinberg configuration 1).

In [16], Vinberg described two the most algebraic K3 surfaces: these are the
K3 surfaces $X_{4}$ and $X_{3}$ of transcendental lattice

$\begin{pmatrix}2&0\\\
0&2\end{pmatrix}\qquad\text{and}\qquad\begin{pmatrix}2&1\\\ 1&2\end{pmatrix},$

respectively. Thanks to results of Shioda and Mitani [10], Shioda and Inose
[9], and to the fact that the class groups of discriminants $-4$ and $-3$ are
trivial, it follows that $X_{4}$ and $X_{3}$ are the unique K3 surfaces of
maximum Picard number and discriminant with minimum absolute value possible.

We start considering the surface $X_{4}$, and we recall how to build a model
for it which is pretty convenient for our purposes. In the complex projective
plane $\mathbb{P}^{2}$, we consider the configuration $\mathcal{L}$ of lines
given by

$\mathcal{L}:\ xyz(x-y)(x-z)(x-y)=0.$

This configuration has three double points and four triple points. By blowing-
up $\mathbb{P}^{2}$ in the four triple points, we obtain a del Pezzo surface
$S$ with a configuration of ten ($-1$)-curves, namely the strict transforms of
the six lines of $\mathcal{L}$ together with the four exceptional divisors.
These ten ($-1$)-curves form a divisor $B$ on $S$, which is simple normal
crossing, with only 15 double points as singularities. After blowing-up these
15 double points, we get a surface $S^{\prime}$, with 15 ($-1$)-curves (the
exceptional divisors) and 10 ($-2$)-curves (the strict transforms of the
irreducible components of $B$, which are now mutually disjoint). By taking a
2:1 cover of $S^{\prime}$ we obtain the K3 surface $X_{4}$, equipped with a
configuration $\mathcal{V}$ of 25 smooth rational curves and 30 double points,
which is described by the Petersen graph in Figure 2.

Figure 2. The Petersen graph.

The 15 edges of the graph correspond to exceptional divisors and the $10$ red
dots correspond to curves from $B$; therefore, $n=25$, $t_{2}=30$ and
$t_{r}=0$ for all $r\geqslant 3$. We now compute the Harbourne constant for
this configuration: we have

$h(X;\mathcal{V})=\frac{(C_{1}+\cdots+C_{25})^{2}-\sum_{j}m_{j}^{2}}{s}=\frac{10-120}{30}\approx-3.666,$

as we somehow expected from the devilish shape of the Petersen graph; together
with the bound in Theorem 2.2, this yields

$-3.7333\approx-112/30\leqslant h(X;\mathcal{V})=-110/30\approx-3.666.$

Vinberg’s $X_{4}$ surface appears also in a different interesting context of
the maximal possible cardinality of a finite complete family of incident
planes in $\mathbb{P}^{5}$ – we refer to [6] for details and results.

######  Example 3.3 (Vinberg configuration 2).

Turning to the K3 surface $X_{3}$, Vinberg [16] provides the reader with a
particularly nice birational model, a complete intersection of a quadric and a
cubic in $\mathbb{P}^{4}$, which we call $Y$:

$\begin{cases}y^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{3}-2(x_{2}x_{3}+x_{1}x_{3}+x_{1}x_{2})\\\
z^{3}=x_{1}x_{2}x_{3}\end{cases}.$

This model contains 9 singular points of type $\rm A_{2}$, namely:

$\displaystyle p_{1}=[0:1:0:1:0],\quad p_{2}=[0:1:0:-1:0],\quad
p_{3}=[0:1:1:0:0],$ $\displaystyle p_{4}=[1:1:0:0:0],\quad p_{5}=[0:0:1:1:0],\
\ \quad p_{6}=[0:0:1:-1:0],$ $\displaystyle p_{7}=[1:0:1:0:0],\quad
p_{8}=[1:0:0:1:0],\ \ \quad p_{9}=[1:0:0:-1:0].$

There are 6 lines lying on $X_{3}$, each of which contains three of the
singular points, in such a way that each singular point is the intersection
point of exactly two of the lines. More precisely, the lines are

$L_{ijk}:\ z=x_{i}=y-(x_{j}-x_{k})=0,$

for any $i,j,k\in\\{1,2,3\\}$, $i\neq j\neq k\neq i$.

The configuration consisting of these 6 lines is shown in Figure 3.

Figure 3. Dual graph of the six lines $L_{ij}$ on $Y$.

We can resolve the singularities of $Y$ by blowing-up twice each singular
point, in order to get a smooth K3 surface, namely $X_{3}$: resolving each
singularity yields two exceptional divisors, which are in fact ($-2$)-curves
as $X_{3}$ is a K3 surface. The exceptional divisors together with the strict
transforms of the six lines on $Y$ yields a new configuration, which we call
$\mathcal{W}$: it consists of $n=6+2\cdot 9=24$ smooth rational curves and it
has only double points as singularities, thus $t_{2}=3\cdot 9=27$ and
$t_{r}=0$ for all $r\geqslant 3$. We now get the Harbourne constant:

$h(X;\mathcal{W})=\frac{(C_{1}+\cdots+C_{24})^{2}-\sum_{r\geqslant
2}r^{2}t_{r}}{s}=\frac{6-27\cdot 4}{27}\approx-3.777,$

and by Theorem 2.2 it follows that

$-3.888\approx-35/9\leqslant h(X;\mathcal{W})=-3.777.$

######  Example 3.4 ($16_{6}$-configuration).

Let $A$ be an abelian surface with an irreducible principal polarization. We
are going to be interested in the singular Kummer surface $K$ given by the
quotient of $A$ by the involution $(-1)_{A}$ (for a detailed account, see [3,
Ch. 10, Sec. 2]). Suppose $L$ is a symmetric line bundle on $X$ defining the
principal polarization; then, the map
$\varphi_{L^{2}}:X\longrightarrow\mathbb{P}^{3}$ defined by the linear system
$|L^{2}|$ factors through an embedding of $K$ in $\mathbb{P}^{3}$. The
singular Kummer surface $K\subset\mathbb{P}^{3}$ has 16 ordinary double points
as singularities, namely the images of the 2-divison points. Moreover, the 16
line bundles algebraically equivalent to $L$ yield 16 planes which are tangent
to $K$ and intersect $K$ along 16 conics (these planes are typically called
tropes). This gives rise to the $16_{6}$ configuration on the Kummer surface:
there are 16 points and 16 planes, each point is contained in exactly 6
planes, and each plane contains exactly 6 points. The points at which each
pair of conics intersects are points of transversal intersection, as the
conics lie in different planes.

Consider the blow-up at the 16 singular points of $K\subset\mathbb{P}^{3}$: as
these are ordinary double points, one blow-up at each point is enough to
resolve the singularities of $K$, and so we obtain a smooth K3 surface $X$.
Since the conics of $K$ intersect transversally, locally over the blown-up
points we get a tree of smooth rational ($-2$)-curves which consists of the
exceptional divisor being intersected by the strict transforms of the six
conics (which are now mutually disjoint).

Consider the configuration $\mathcal{C}=\\{C_{1},\dots,C_{32}\\}$ of
($-2$)-curves on $X$ consisting of the 16 exceptional divisors and the strict
transforms of the 16 conics on $K$: these curves only meet in double points
because we have blown-up all the intersection points of the conics, and the
number of double points is exactly $6\cdot 16=96$. Therefore, for the
configuration $\mathcal{C}$ we have $n=32$, $t_{2}=96$ and $t_{r}=0$ for
$r\geqslant 3$. The Harbourne constant is then

$h(X;\mathcal{C})=\frac{(C_{1}+\cdots+C_{32})^{2}-\sum_{r\geqslant
2}r^{2}t_{r}}{s}=-8/3\approx-2.666,$

and together with the lower bound of Theorem 2.2 this shows that

$-3.08333\approx-296/96\leqslant h(X;\mathcal{C})=-8/3\approx-2.666.$

######  Example 3.5 (Schur quartic surface).

Let $S$ be the quartic surface in $\mathbb{P}^{3}$ given by

$S:\ x^{4}-xy^{3}=z^{4}-zw^{3}.$

The surface $S$ is called the Schur quartic surface, and it is the surface
that achieves the upper bound of 64 lines for quartic surfaces (see, for
example, [15]). The 64 lines on $S$ are divided into two classes, namely lines
of the 1st kind and of the 2nd kind. Lines of different kind can be
distinguished according to the singular fibers of the fibration they induce on
$S$; the singular fibers of an elliptic fibration induced by a line of the 1st
or 2nd kind are depictued in Figures 4 and 5

$\times 4$$\times 6$ Figure 4. Singular fibers induced by a line of 1st kind.
$\times 2$$\times 4$$\ell$ Figure 5. Singular fibers induced by a line of 2nd
kind.

The configuration $\mathcal{S}$ of lines on $S$ counts 64 lines, 8 quadruple
points, 64 triple points and 336 double points. We can extract a
subconfiguration $\check{\mathcal{S}}$ of $\mathcal{S}$, which is obtained by
only considering the lines of the 2nd kind: this configuration consists of 16
lines and only 8 quadruple points. We can now compute the Harbourne constant
in this case:

$h(S;\check{\mathcal{S}})=-8.$

The lower bound given by Theorem 2.2 finally yields

$-9\leqslant h(S;\check{\mathcal{S}})=-8.$

It is interesting to notice that the same is achieved by means of the Bauer
configuration of lines on the Fermat quartic surface

$F:\ x^{4}+y^{4}+z^{4}+w^{4}=0,$

as it is shown in [14, Example 4.3]. However, we remark that, in the case of
the Fermat surface, all lines are of the 1st kind.

######  Example 3.6 (Double Kummer pencil).

Let $E$ and $E^{\prime}$ be two elliptic curves. Recall that any elliptic
curve is a 2:1 cover of $\mathbb{P}^{1}$ ramified at 4 points, and that the 4
ramification points are exactly the 2-torsion points of the elliptic curve (to
see this, work with an elliptic curve in Legendre form). Consider the product
(abelian) surface $E\times E^{\prime}$, which comes with two projections onto
the factors. We can see a configuration $\mathcal{E}$ of 8 elliptic curves on
$E\times E^{\prime}$: these are the fibers of $p$ over the 2-torsion points of
$E$, which we call $C_{i}$ ($1\leqslant i\leqslant 4$), together with the
fibers of $p^{\prime}$ over the 2-torsion points of $E^{\prime}$, denoted by
$D_{j}$ ($1\leqslant j\leqslant 4$). Each $C_{i}$ intersects all $D_{j}$’s,
and viceversa, thus ${\rm Sing}(\mathcal{E})$ consists of 16 points, which in
turn are the 2-torsion points of $E\times E^{\prime}$. We can now consider the
K3 surface ${\rm Km}(E\times E^{\prime})$, the Kummer surface of $E\times
E^{\prime}$, obtain by first quotienting by the action of $(-1)_{E\times
E^{\prime}}$ and then resolving the 16 singularities of type $\rm A_{1}$. The
configuration $\mathcal{E}$ yields a configuration $\mathcal{K}$ of
($-2$)-curves on ${\rm Km}(E\times E^{\prime})$, which consists of the images
of the curves $C_{i}$ and $D_{j}$ ($1\leqslant i,j\leqslant 4$) in ${\rm
Km}(E\times E^{\prime})$ and the 16 exceptional divisors (with their reduced
structure). The configuration $\mathcal{K}$ is the double Kummer pencil
configuration, and it consists of $n=24$ ($-2$)-curves intersecting only at
$t_{2}=2\cdot 16=32$ double points (see Figure 6).

Figure 6. The double Kummer pencil configuration.

This yields the following Harbourne constant:

$h({\rm Km}(E\times E^{\prime}),\mathcal{K})=-14/4\approx-3.5,$

which combined with Theorem 2.2 results in

$-3.75=-15/4\leqslant h(X;\mathcal{K})=-14/4=-3.5.$

######  Example 3.7 (Enriques surfaces covered by symmetric quartic surfaces).

This example is borrowed from a recent paper of Mukai and Ohashi [13]. Let
$\bar{X}$ be the quartic in $\mathbb{P}^{3}$ given as the zero locus of

$\bar{X}:\ \Big{(}\sum_{i<j}x_{i}x_{j}\Big{)}^{2}=kx_{0}x_{1}x_{2}x_{3}.$

This is a singular hypersurface, with four singularities of type $\rm D_{4}$,
namely the vertices of the fundamental tetrahedron. The coordinate planes cut
$\bar{X}$ in conics with multiplicity two, which are also called tropes, and
each one of these conics passes through 3 of the singular points. After
resolving the $\rm D_{4}$-singularities, we obtain a K3 surface $X$, which is
equipped with an interesting configuration $\mathcal{C}$ of ($-2$)-curves,
namely the exceptional divisors coming from the resolution of the
singularities and the strict transforms of the tropes. The configuration
$\mathcal{C}$ is described by the dual graph in Figure 7, where all the
intersections are points of multiplicity two, thus $n=20$, $t_{2}=24$ and
$t_{r}=0$ for all $r\geqslant 3$.

Figure 7. The configuration $\mathcal{C}$ of smooth rational curves on $X$.

We can compute the Harbourne constant for this configuration:

$h(X;\mathcal{C})=-11/3\approx 3.666,$

and thanks to Theorem 2.2 we also see that

$-4.333\approx-13/3\leqslant h(X;\mathcal{C})=-11/3\approx 3.666.$

From the K3 surface $X$, we can construct an Enriques surface with an
interesting configuration of smooth rational curves. The singular surface
$\bar{X}$ is endowed with the standard Cremona transformation

$\varepsilon:[x_{0}:x_{1}:x_{2}:x_{3}]\longmapsto[x_{0}^{-1}:x_{1}^{-1}:x_{2}^{-1}:x_{3}^{-1}],$

which extends to a morphism on the blown-up surface $X$. For general values of
$k$ (precise conditions are given in [13, pag. 1]), there are no fixed points
of $\varepsilon$ on $X$, and thus the quotient $X/\varepsilon=:S$ is an
Enriques surface. The morphism $\varepsilon$ acts on the cube-shaped diagram
in Figure 7 by point symmetry (i.e. symmetry with respect to the center of the
cube), and thus the quotient diagram is the tetrahedron graph in Figure 8,
also known as the 10A configuration in Mukai-Ohashi’s notation.

Figure 8. The $10A$ configuration on the Enriques surface $S$.

The graph describes the interaction of the images of the rational curves on
$X$ modulo quotient by $\varepsilon$. We now compute the Harbourne constant
for such a configuration of curves:

$h(X;10A)=-11/3\approx 3.666.$

For Enriques surfaces, the bound in Theorem 2.2 takes the stronger form

$h(X;10A)\geqslant-4+\frac{2n+t_{2}-36}{s}=-13/3,$

and thus

$-4.333\approx-13/3\leqslant h(X;10A)=-11/3\approx 3.666.$

######  Example 3.8 (A Hessian K3 surface and its Enriques quotient).

The last example we would like to present uses the construction of the Hessian
K3 surface associated to a cubic surface in $\mathbb{P}^{3}$; details can be
found in [4]. Let $S$ be a smooth cubic surface defined by the Sylvester form

$S:\qquad\sum_{i=0}^{4}x_{1}^{3}=\sum_{i=0}^{4}x_{i}=0.$

The union of the five planes in $\mathbb{P}^{3}$ defined by $x_{i}=0$ is
called the pentahedron of $S$. The $10$ edges $L_{ijk}$ of the pentahedron are
lines on $S$. We can consider the Hessian $Y$ of $S$: it is the (singular)
surface defined by

$Y:\qquad(x_{0}x_{1}x_{2}x_{3}x_{4})\sum_{i=0}^{4}\frac{1}{x_{i}}=\sum_{i=0}^{4}x_{i}=0.$

The ten lines $L_{ij}$ lie on $Y$, and the vertices $P_{ijk}$ of the
pentahedron are the singular points of $Y$ (double points). The
desingularization $X$ of $Y$ is a K3 surface, called the Hessian K3 surface
associate to $S$. There are $20$ rational curves on $X$, $10$ of which are the
strict transforms of the $L_{ij}$’s, and we will call them by $N_{ij}$. The
remaining ones are the curves arising from the resolution of the singularities
at the $P_{ijk}$’s, and they will be denoted by $N_{ijk}$.

We can find $20$ more rational curves on $X$ by looking at the Eckardt points
of $S$. A smooth cubic surface has $27$ lines and $45$ plane sections which
are unions of three lines. In case three coplanar lines meet at a single
point, this point is called an _Eckardt point_. Each Eckardt point on $S$
yields a pair of lines on $Y$ meeting at one of the $P_{ijk}$’s, for a total
of $20$ lines. The strict transforms of these extra lines yield $20$ new
rational curves on $X$. We can read off the intersection numbers of all these
curves from [4, Sections 1-2]: the configuration $\mathcal{C}$ given by the
$40$ aforementioned rational curves has $130$ double points. The Harbourne
constant is

$h(X;\mathcal{C})=\frac{180-520}{130}=-34/13\approx-2.615384;$

combining with Theorem 2.2, we get

$-2.9384615\approx-4+\frac{138}{130}\leqslant
h(X;\mathcal{C})=-34/13\approx-2.615384.$

From $X$, we can cook up an Enriques surface $\bar{X}$: every Hessian quartic
surface is equipped with a birational involution, which becomes a fixed-point-
free morphism on the Hessian K3 surface, yielding an Enriques surface by
taking the quotient [5]. Consider $X$ as above, equipped with its Enriques
involution $\tau$: this automorphism swaps $N_{ijk}$ and $N_{lm}$ (for
$\\{i,j,k,l,m\\}=\\{0,1,2,3,4\\}$), and it also swaps the two rational curves
arising from each Eckardt point. As none of the curves in $\mathcal{C}$ is
fixed by $\tau$, on the Enriques quotient $\bar{X}$, we obtain a configuration
$\bar{\mathcal{C}}$ of 20 rational curves, meeting at 65 double points. As the
local intersections are preserved, the Harbourne constant and its lower bound
remain unchanged:

$-2.9384615\approx-4+\frac{69}{65}\leqslant
h(\bar{X};\bar{\mathcal{C}})=-34/13\approx-2.615384.$

### _Acknowledgement._

The present paper has grown out from discussions the authors had during the
time when Piotr Pokora was visiting Roberto Laface at the Leibniz Universität
Hannover. It is a great pleasure to thank the Institut für Algebraische
Geometrie for generous fundings and excellent working conditions. The authors
would like to express their gratitude to Klaus Hulek and Matthias Schütt for
stimulating discussions about the topic, Igor Dolgachev for pointing out
Example 3.8, and Bert van Geemen for sharing his insights on the subject. The
first-named author would like to thank Davide Cesare Veniani for sharing his
insights on some of the examples. The second author would like to express his
gratitude to Łukasz Sienkiewicz for pointing out Vinberg’s paper [16] and his
construction, and to Stefan Müller-Stach for useful comments. The second
author is partially supported by National Science Centre Poland Grant
2014/15/N/ST1/02102 and SFB 45 _Periods, moduli spaces and arithmetic of
algebraic varieties_.

## References

  * [1] Th. Bauer & B. Harbourne & A. L. Knutsen & A. Küronya & S. Müller–Stach & X. Roulleau & T. Szemberg: Negative curves on algebraic surfaces. Duke Math. J. 162: 1877 – 1894 (2013).
  * [2] Th. Bauer & S. Di Rocco & B. Harbourne & J. Huizenga & A. Lundman & P. Pokora & T. Szemberg: Bounded Negativity and Arrangements of Lines, International Mathematical Research Notices vol. 2015: 9456 - 9471 (2015), doi:10.1093/imrn/RNU236.
  * [3] C. Birkenhake & H. Lange: Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften, 302, Springer-Verlag, Berlin (2004).
  * [4] E. Dardanelli & B. van Geemen: Hessians and the moduli space of cubic surfaces. Comtemp. Math. 422: 17 – 36 (2007).
  * [5] I. Dolgachev & J. Keum: Birational automorphisms of quartic Hessian surfaces. Trans. Amer. Math. Soc. 354(8): 3031 – 3057 (2002).
  * [6] M. Donten-Bury & B. van Geemen & G. Kapustka & M. Kapustka & J. A. Wiśniewski: A very special EPW sextic and two IHS fourfolds, arXiv:1509.06214.
  * [7] B. Harbourne: Global aspects of the geometry of surfaces. Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica vol. IX: 5 – 41 (2010).
  * [8] F. Hirzebruch: Arrangement of lines and Algebraic surfaces. Arithmetic and geometry, Vol. II, 113–140, Progr. Math., 36, Birkhäuser, Boston, Mass., 1983.
  * [9] H. Inose & T. Shioda: On singular K3 surfaces, Complex analysis and algebraic geometry, pp. 119–136. Iwanami Shoten, Tokyo (1977).
  * [10] N. Mitani & T. Shioda: Singular abelian surfaces and binary quadratic forms, Classification of algebraic varieties and compact complex manifolds, pp. 259 – 287. Lecture Notes in Math., Vol. 412, Springer, Berlin (1974).
  * [11] Y. Miyaoka: On the Chern numbers of surfaces of general type. Invent. Math. 42(1): 225 – 237 (1977).
  * [12] Y. Miyaoka: The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268: 159 – 171 (1984).
  * [13] S. Mukai & H. Ohashi: The automorphism group of Enriques surfaces covered by symmetric quartic surfaces, Recent advances in algebraic geometry, 307–320, London Math. Soc. Lecture Note Ser., 417, Cambridge Univ. Press, Cambridge (2015).
  * [14] P. Pokora: Harbourne constants and arrangements of lines on smooth hypersurfaces in $\mathbb{P}^{3}_{\mathbb{C}}$, to appear in Taiwanese J. Math., arXiv:1505.03822, doi:10.11650/tjm.19.2015.6338.
  * [15] S. Rams & M. Schütt: 64 lines on smooth quartic surfaces, Math. Ann. 362 no. 1-2: 679 – 698 (2015).
  * [16] E. B. Vinberg: The two most algebraic K3 surfaces, Math. Ann. 265: 1 – 21 (1983).