\item $\indeg (\Der_k(R)/\m^{-1} \varepsilon) \le 2d -5-a(R)\, ;$
\item
$\findeg (\Der_k(R)/R \varepsilon) \le d-2$  if  ${\rm char} \, k =0$  and  $\mathcal C$  is smooth and arithmetically
\end{enumerate}
\end{theorem}
\begin{proof} We may assume that $k$ is infinite and $n\ge3$. Let $x_1, x_2, x_3$ be general linear forms in $R$ and consider the subalgebra $A=k[x_1,x_2,x_3]$ of $R$. Notice that $A\subset R$ is a finite and birational extension by \autoref{UV} and that $A$ is a hypersurface ring.  Thus \autoref{UB}(d) gives
\[ \findeg (\Der_k(A)/A\varepsilon_A ) \leq a(A) +1 \, .\]
Also observe that  $e(A)=e(R)$ by \autoref{UV}, hence $a(A)=e(A)-3=e(R)-3=d-3$.

Now part (a) follows because
\[\indeg (\Der_k(R)/\m^{-1} \varepsilon_R) \leq \findeg (\Der_k(A)/A\varepsilon_A ) + a(A) -a(R)\, \]
by \autoref{GP}.
If the assumptions of part (b) are satisfied, then  $R$ is the integral closure of $A$ and $R$ is a domain. Hence every derivation of $A$ can be
extended to a derivation of $R$, according to \cite{Seidenberg}*{Theorem, page 168}. From \autoref{Dsed} in the proof of
\autoref{GP} we see that there are embeddings
\begin{tikzcd}
\Der_k(A)/A\varepsilon_A   \arrow[hookrightarrow]{r}
& \Der_k(A,R)/\m^{-1}\varepsilon_A   \arrow[r, hookleftarrow, "\varphi"]  & \Der_k(R)/\m^{-1}\varepsilon_R\, .
\end{tikzcd}
Since every derivation of $A$ can be extended to a derivation of $R$, the map $\varphi$ is an isomorphism. Thus we obtain an embedding $\, \Der_k(A)/A\varepsilon_A  \hookrightarrow \Der_k(R)/\m^{-1} \varepsilon_R \,$. As $\depth R >0 $, this embedding shows that
\[\findeg (\Der_k(R)/\m^{-1} \varepsilon_R) \leq \findeg (\Der_k(A)/A\varepsilon_A )\, ,\]
which proves part (b).
\end{proof}

\vspace{.0000001cm}

\begin{proposition} Let $k$ be a perfect field and $\C\subset \mathbb P_k^{n-1}$ be a reduced curve that is arithmetically Cohen-Macaulay. Let $R$ be the homogenous coordinate ring of $\C$. One has
\[
\findeg (\Der_k(R)/R\varepsilon) \le \max\{ r(R)\cdot (a(R)+2)-2\, , \, a(R)+1\}\, .\]

\end{proposition}
\begin{proof} In view of \autoref{UB}(b) it suffices to prove that $\findeg \omega_R^* \le r(R) \cdot (a(R)+2)-2$. We may assume that $n\ge 3$ and that $\C\subset \mathbb P_k^{n-1}$  is non-degenerate. We write $S$ for the coordinate ring of $ \mathbb P_k^{n-1}$ and consider a minimal homogeneous free $S$-resolution  $\mathbb F_{\bullet}$ of $R$.   Since $R$ is Cohen-Macaulay, the resolution  $\mathbb F_{\bullet}$ has length $n-2$ and $F_{n-2}$ is generated in degrees at most $a(R)+n$. Moreover, $\indeg F_{n-3} \ge n-2$ because the curve is non-degenerate. It follows that the entries of $\varphi$, the last matrix in the resolution $\mathbb F_{\bullet}$, have degrees at most $a(R)+2$.

Let $\alpha$ be a general homogeneous minimal generator of $\omega_R$ of maximal degree. Observe that ${\rm ann}_R\, \alpha=0$. Moreover, the graded module
$\omega_R/R \alpha$ is minimally generated by $r(R)-1$ homogeneous elements and is presented by the transpose of $\varphi$, with one row removed.
This is a matrix with $r(R)-1$ rows and homogeneous entries of degrees at most $a(R)+2$. The ideal $\a$ of $(r(R) -1) \times (r(R)-1)$ minors of this matrix
satisfies $\a \subset \ann_R(\omega_R/R\alpha) \subset {\sqrt \a}$, has positive grade, and is generated by forms of degrees at most $(r(R)-1)(a(R)+2)$.
Thus, there exists a homogeneous non-zerodivisor $b \in \a \subset \ann_R(\omega_R/R\alpha)$ with $\deg b \leq (r(R)-1)(a(R)+2)$.

Now the exact sequence
$$ 0 \longrightarrow \omega_R^* \longrightarrow (R\alpha)^* \longrightarrow \Ext^1_R(\omega_R/R\alpha, R) $$
shows that
$$b (R \alpha)^* \subset \omega_R^* \, .$$
Since $(R \alpha)^* \cong R(\deg \alpha)$ and $\deg \alpha \geq  \indeg \omega_R = -a(R)$, it follows that
$\findeg (R\alpha)^* \leq a(R)$. As moreover $b$ is a non-zerodivisor, we conclude that
$$\findeg \omega_R^* \leq \findeg b (R\alpha)^* = \deg b + \findeg (R \alpha)^* \leq (r(R)-1)(a(R)+2)+a(R) \, ,$$
as required.
%because $b$ is a non-zerodivisor.
\end{proof}

We finish this section by providing the minimal graded free resolution of the module $\Der_k (R)/R\varepsilon$ for the case of a smooth arithmetically Cohen-Macaulay curve in $\mathbb P_k^3$. From this we obtain, for instance, the initial degree, the minimal number of generators, and the entire Hilbert series of  $\Der_k (R)/R\varepsilon $. In particular, we see that the upper bound of \autoref{UB}(d) fails dramatically without the  nearly Gorenstein assumption.

\begin{theorem}\label{curvein3} Let $k$ be a perfect field and $\C\subset \mathbb P_k^{3}$ be a curve of degree $d$ that is smooth and arithmetically Cohen-Macaulay. Let $R$ be the homogenous coordinate ring of $\C$ and $S=k[x_1, \ldots, x_4]$  be the homogenous coordinate ring of $\mathbb P_k^{3}$. Let
%Write $R=S/I$, where $S=k[x_1, \ldots, x_4]$ is a polynomial ring and $I$ is a homogenous ideal of $S$. Let
$$F_{\bullet}: 0 \lto F_2=\oplus_{j=1}^{n-1} S(-b_j) \stackrel{\varphi}{\lto} F_1=\oplus_{i=1}^{n} S(-a_i) \lto S$$
be the minimal homogenous $S$-free resolution of $R$. We may assume that $a_1\le \ldots \le a_n$.
\begin{enumerate}[$(a)$]
\item
\[\Der_k(R)/R\varepsilon\cong  \begin{cases}\m (-a(R))& \mbox{if } n=2  \mbox{ and d is  not a multiple of the characteristic} \\
\omega_R^*& n\ge 3 \, .\end{cases}
\]
\item
\[\indeg (\Der_k(R)/R\varepsilon)=  \begin{cases}a_1+a_2-3 & \mbox{if } n=2  \mbox{ and d is  not a multiple of the characteristic} \\
a_1+a_2-4& n\ge 3 \, .\end{cases}
\]
\item If $n \ge 3$, then the minimal homogenous $S$-free resolution of $\, \Der_k (R)/R \varepsilon$ is of the form
\[
\begin{tikzcd}[row sep={-1.3em}]
&  \bigoplus\limits_{2\le j_1\le j_2 \le n-1} S(-b_{j_1}-b_{j_2}+4) &  \bigoplus\limits_{\substack{2\le j\le n-1 \\ 1\le i\le n}} S(-b_j-a_i+4)  & \\
0 \arrow[r, shorten >=8ex]   &\quad  \bigoplus  \arrow[r,  shorten >=10ex] \qquad \qquad  & \qquad \quad  \bigoplus  \arrow[r,  shorten >=3ex] \qquad \qquad & \bigoplus\limits_{1\le i_1 < i_2\le n} S(-a_{i_1}-a_{i_2}+4)\\
& \bigoplus\limits_{1\le j\le n-1} S(-b_1-b_j+4) & \bigoplus\limits_{1\le i\le n} S(-b_1-a_i+4)  &
\end{tikzcd}
\]
\item Assume $n\ge 3$ and let $\psi$ be the $n-2$ by $n$ matrix obtained by deleting the first column of $\varphi$. One has
%After adding suitable multiples of the first columns to the other columns we may assume that the ideal of maximal minors of the matrix $\psi$ consisting of the last $n-2$ columns of $\varphi$ as maximal height, namely
$\htt I_{n-2}(\psi)=3$ and
$$\Der_k(R)/R\varepsilon \cong \omega_R^*\cong \frac{I_{n-2}(\psi)}{I_{n-1}(\varphi)} (4-b_1)\, .
\item Assume $n\ge 3$. Write $F_2=F_{21} \oplus F_{22}$ where $F_{21}$  is generated by the first basis element of $F_2$ and $F_{22}$ is generated by the remaining basis elements, so that $\psi: F_{22}\lto F_1, $ and let $\pi: F_2 \twoheadrightarrow F_{22}$ be the natural projection. Write $-^{\vee}=\Hom_S(-, S)\, .$ Set $b=\sum\limits_{j=1}^{n-1} b_j$ and $b'=\sum\limits_{j=2}^{n-1}b_j.$ Consider the diagram
\begin{tikzcd}[row sep={4em}]
0\arrow{r} & D_2(F_{22}) \otimes \bigwedge\limits^n F_1^{\vee}(-b') \arrow{r} &F_{22} \otimes \bigwedge\limits^{n-1} F_1^{\vee}(-b') \arrow{r} & \bigwedge\limits^{n-2} F_1^{\vee}(-b') \\
&  \hspace{3cm} 0   \arrow[r]  & F_2 \otimes \bigwedge\limits^n F_1^{\vee} (-b)\arrow{r} \arrow[u, "-\pi \otimes \delta"] & \bigwedge\limits^{n-1} F_1^{\vee}(-b) \arrow[u, "\delta"]  \, ,
\end{tikzcd}
where the first and the second row are the truncated Eagon-Northcott complexes associated to the matrices $\psi$ and $\varphi$, respectively, and $\delta$ is the differential in the Koszul complex of the sequence consisting of the entries of the first column of $\varphi$. These vertical maps give a morphism of complexes $u_{\bullet}$, and the mapping cone $C(u_{\bullet})$ is a minimal homogeneous $S$-free resolution of $\Der_k (R)/R\varepsilon.$
\end{enumerate}
\end{theorem}
\begin{proof} We first prove the claim about the height and the second isomorphism in part (d). Notice that
$$F_{\bullet}^{\vee}: 0 \lto S(-4) \lto  F_1^{\vee}(-4) \stackrel{\varphi^{\vee}}{\lto}  F_2^{\vee}(-4)=\oplus_{j=1}^{n-1} S(b_j-4) $$
is a minimal homogeneous free $S$-resolution of $\omega_R$. Since $\omega_R$ is a torsionfree $R$-module and $R$ is a domain, the image $\alpha \in \omega_R$ of the first basis element of $F_2^{\vee}(-4)$  generates a submodule $R\alpha \cong  R(b_1-4).$ Notice that $\dim ( \omega_R/R\alpha ) \le 1.$ As an $S$-module, $ \omega_R/R\alpha $ is presented by the $n$ by $n-2$ matrix $\psi^{\vee}.$ It follows that $\htt I_{n-2} (\psi)=\htt I_{n-2} (\psi^{\vee}) \ge 3.$ Therefore  $\ann_S (\omega_R/R\alpha) =I_{n-2}(\psi^{\vee})=I_{n-2}(\psi)$
according to \cite{BEann}*{Theorem page 232}.

On the other hand, a shift of $\omega_R$ is isomorphic to a homogenous ideal $K$ of $R$. Let $\beta \in K$ be  the element corresponding to $\alpha.$ One has
\begin{eqnarray*}
\omega_R^* (b_1-4)&\cong& \Hom(\omega R, R\alpha)\cong \Hom(K, R\beta)\cong R\beta:_R K  \\
&=&\ann_R (K/R\beta) =\ann_R (\omega_R/R\alpha)=\frac{\ann_S (\omega_R/R\alpha)}{I}
\\
&=&\frac{I_{n-2}(\psi)}{I_{n-1}(\varphi)}\, .
\end{eqnarray*}

Next we prove parts (a) and (b), which will also completes the proof of (d). If $n=2$, the assertions follow from \autoref{SmoothGOR}. Hence we may assume that $n\ge 3$. The second isomorphism in (d) shows that $\indeg \omega_R^*=\indeg I_{n-2}(\psi)+b_1-4.$ On the other hand, $\indeg I_{n-2}(\psi)=\sum\limits_{j=2}^{n-1} b_j -\sum\limits_{i=2}^{n}a_i =a_1+a_2-b_1$. The last equality holds because $\sum\limits_{j=1}^{n-1} b_j =\sum\limits_{i=1}^{n}a_i,$ by the Hilbert-Burch theorem. We conclude that $\indeg \omega_R^*=a_1+a_2-4.$ Now parts (a) and (b) follow from \autoref{anticanonicalM} once we have shown that $a_1+a_2-4>a(R).$

To this end we may assume that  $b_1\le \ldots \le b_{n-1}.$ Hence $a(R)=b_{n-1}-4$ and we need to prove that $a_1+a_2>b_{n-1}$. We consider the degree matrix associated to $\varphi$, which is the $n-1$ by $n$ matrix with entries $u_{ij}=b_j-a_i.$ Notice that $\varphi_{ij}=0$ if $u_{ij}\le 0$. It easily follows that $u_{j+2,j}>0$ for all $j$ since $I$ is a prime ideal (see also \cite{GM89}*{page 3142}). As $a_2=u_{1,n-1}+ \sum\limits_{j=1}^{n-2} u_{j+2,j}$ and $n\ge 3,$ we see that $a_2>b_{n-1}-a_1$.

We now prove (e). Since $\htt I_{n-2}(\psi)\ge 3$ by part (d), the two truncated Eagon-Northcott complexes are minimal homogeneous   $S$-free resolutions of $I_{n-2}(\psi)$ and $I_{n-1}(\varphi)$, respectively. One easily checks that $u_{\bullet}$ is a morphism of complexes. Thus $C(u_{\bullet})$ is a homogeneous $S$-free resolution of $\frac{I_{n-2}(\psi)}{I_{n-1}(\varphi)}.$ It is minimal because the matrices of the vertical maps have entries in $\m_S.$ We deduce part (c) from (e), repeatedly using the equality $\sum\limits_{j=1}^{n-1} b_j =\sum\limits_{i=1}^{n}a_i.$
\end{proof}

\medskip

\section{The Euler derivation in the module of derivation}\label{secEuler}

%In the next corollary and its proof, we use the notation introduced at the beginning of Section 2.

In proving the graded case of the Zariski-Lipman conjecture, Hochster showed that, for any Noetherian positively graded  algebra $R$ over a field of characteristic zero, the Euler derivation is  a minimal generator of $\Der_k(R)$, unless $R$ is a polynomial ring over a subalgebra \cite{Hoc}*{pg 412}. One may wonder whether the Euler derivation can generate a free direct summand. In this section we use the results from \autoref{SecEK} to address this issue and the related question of whether the natural map  $\, \Der_k(R) \to L^*$ of \autoref{1.2} can be surjective.

\smallskip

\begin{proposition}\label{splitting} Let $R$ be a two-dimensional Noetherian standard graded algebra over a field $k$, with homogeneous maximal ideal $\m$, and assume that the multiplicity of $R$ is not a multiple of the characteristic of $k$. If $R$ is Gorenstein, then the natural map $\, \Der_k(R) \to L^*$ is not surjective and the Euler derivation does not generate a free direct summand of $\ \Der_k(R)$.
\end{proposition}
\begin{proof} Let $S$ be a polynomial ring over $k$ of dimension $\ge 3$ with homogeneous maximal ideal $\m_S$ that maps homogeneously onto $R$. The commutative diagram with exact rows
$$\begin{tikzcd} 0 \arrow{r} &Z \arrow{r} \arrow{d}& \Omega_k(S)  \arrow{d}\arrow{r} &\m_S \arrow{d} \arrow{r} &0 \\
0 \arrow{r} &L \arrow{r}  & \Omega_k(R) \arrow{r} &\m \arrow{r} &0
\end{tikzcd} $$
induces a commutative diagram
$$\begin{tikzcd} H^1_{\m_S}(\m_S) \arrow{r}{\alpha} \arrow{d}&  H^2_{\m_S}(Z)  \arrow{d}{\beta}  \hphantom{ \ .}\\
H^1_{\m}(\m) \arrow{r}{\gamma}  &  H^2_{\m}(L) \ .
\end{tikzcd} $$
Here $\alpha$ is an isomorphism since ${\rm depth}_{\m_S} \,  \Omega_k(S)\ge 3$, and $\beta$ is nonzero by \autoref{Th4.3}. We conclude that $\gamma$ is nonzero.

We use the exact sequence of \autoref{1.2}
$$\begin{tikzcd}0 \arrow{r} &\Der_k(R)/R\, \varepsilon \arrow{r} {\delta} & L^* \arrow{r}{\nu} &\Ext^2_R(k,R)   \cong \Ext^1_R(\m,R)
\end{tikzcd}
that was also induced by the
%On the other hand, dualizing the
exact sequence
$ \ 0 \to L \to  \Omega_k(R)\to \m \to 0     \      . $
% into $R$, \textcolor{blue}{(and since $R$ is Cohen--Macaulay)}, we obtain
%$$\begin{tikzcd}0 \arrow{r} &\Der_k(R)/R\, \varepsilon \arrow{r} {\delta} & L^* \arrow{r}{\nu} &\Ext^1_R(\m,R)\cong \Ext^2_R(k,R)
% \end{tikzcd} $$
% (see also \autoref{1.2}).
Since $R$ is Gorenstein and $\gamma\not=0$, local duality shows that $\nu\not=0$. Thus $\delta$ is not surjective. Moreover, ${\rm depth} \Der_k(R)/R \, \varepsilon=1$ by the depth lemma because ${\rm im}\, \nu$ has depth zero. Thus $R \, \varepsilon$ cannot be a direct summand of $\Der_k(R)$, a module of depth two.
\end{proof}

Surprisingly, if $R$ is Cohen--Macaulay, but not Gorenstein then the natural map $\Der_k(R)\to L^*$ can be surjective in dimension two. This is always the case for the coordinate rings of rational normal curves of degree $n\ge 3$ in $\mathbb P^n_k,$ as we will see in \autoref{scrolls} below.

\begin{lemma}\label{lemmascroll} Let $T$ be a standard graded Noetherian domain with ${\rm grade} \, T_{+}\ge 2$. Let $s \in \mathbb N$ be invertible in $T$ and denote the    $s$th Veronese functor by $\--^{(s)}.$ Then
$$ {\rm Der}_{T_0}(T^{(s)})=({\rm Der}_{T_0}(T))^{(s)}\, .
\end{lemma}
\begin{proof}We may assume that $T_+\not=0$ and $s\geq 2$. We write $R=T^{(s)}$ and
consider the exact sequence
\begin{equation}\label{canonicalseq}\begin{tikzcd} T\otimes_R \Omega_{T_0}(R) \arrow[r, "\varphi"] &\Omega_{T_0}(T) \arrow{r}  &  \Omega_R(T) \arrow{r} & 0
\end{tikzcd}
\end{equation}
We first prove that $\Supp(\Omega_R(T))\subset V(T_{+}).$ Let $d:T\to \Omega_{T_0}(T)$ denote the universal derivation. Let $\p\in \Spec(T)\setminus V(T_{+})$
and  $\ell$  an arbitrary linear form in $T$. We need to show that $d(\ell) \in (\im\varphi)_{\p}.$ We choose $x\in T_1\setminus \p.$  Since $s\,x^{s-1}d(x)=d(x^s)\in \im\varphi$ it follows that $d(x)\in (\im\varphi)_{\p}.$
Now the containment $(s-1)\,x^{s-2}\ell \, d(x)+x^{s-1}d(\ell)=d(\ell x^{s-1})\in \im \varphi \,$ implies that $d(\ell)\in (\im \varphi)_{\p}.$

Let $K$ and $L$ be the quotient fields of $R$ and $T$, respectively. Since $\Omega_K(L)=0$ by the above, the field extension $K \subset L$ is separable algebraic. Therefore $\varphi \otimes_T L$ is an isomorphism, which shows that ${\rm ker} \, \varphi$ is a torsion module. Again, since  $\Supp(\Omega_R(T))\subset V(T_{+})$, it follows that ${\rm grade}\, \Omega_R(T) \ge 2, $ hence ${\rm Ext}^1_T(\Omega_R(T), T)=0$. Now, dualizing the sequence \autoref{canonicalseq} into $T$ gives the identification
$$ {\rm Der}_{T_0}(T)= {\rm Der}_{T_0}(R,T)\, .

Thus $\,  {\rm Der}_{T_0}(R)\subset  {\rm Der}_{T_0}(T),$ and a degree argument immediately yields the desired equality.
%$\,  {\rm Der}_{T_0}(T^{(n)})=({\rm Der}_{T_0}(T))^{(n)}\, .$
\end{proof}

\begin{corollary}\label{corscrolls}
Let $T=T_0[z_1, \ldots, z_t]$ be a standard graded polynomial ring with $t\ge 2$,  and let $s \ge 2$ be invertible in $T$. Then
$ {\rm Der}_{T_0}(T^{(s)})$ is the $T^{(s)}$-submodule of $\ \Der_{T_0}(T)=\oplus_{i=1}^{t} \ T \, \frac{\partial}{\partial z_i}$ minimally generated by the  homogeneous elements
$ z_i \frac{\partial}{\partial z_j}$ for $1\le i, j\le t$.
\end{corollary}
\begin{proof} Applying \autoref{lemmascroll} we obtain
\[ {\rm Der}_{T_0}(T^{(s)})=({\rm Der}_{T_0}(T))^{(s)}=T^{(s)} [{\rm Der}_{T_0}(T)]_0\, ,
\]
where the last equality holds because ${\rm Der}_{T_0}(T)$ is generated in degree $-1$ and $-s<-1\le 0$.
\end{proof}

The coordinate ring of the rational normal curve of degree $n$ in $\mathbb P^n_k$ is of the form $R=S/I$, where $S=k[x_0, \ldots, x_n]$ and $I$ is the ideal generated
by the maximal minors of the matrix
\begin{equation}\label{scrollM}
\begin{bmatrix}
x_{0}       & x_{1} & x_{2} & \dots & \dots & x_{n-1} \\
x_{1}       & x_{2} &  \dots &  \dots& x_{n-1} & x_{n}
\end{bmatrix}\, .
\end{equation}
\vspace{0.01in}

\noindent
We write $y_i$ for the images of $x_i$ in $R$.

\begin{proposition}\label{scrolls} Let $R$ be the coordinate ring of the rational normal curve of degree $n\ge 3$ in $\mathbb P^n_k,\ $ and assume that the characteristic of the field $k$ is not a divisor of $n$.
\begin{enumerate}[$($a$)$]
\item The module $\Der_k(R)$ is the $R$-submodule of $\ \Der_k(S,R)=\oplus_{i=0}^{n} \ R \, \frac{\partial}{\partial x_i}$ minimally generated by the following  $4$ homogeneous elements of degree $0$,
$$ \sum_{i=0}^{n-1}\, (n-i)\, y_i \, \frac{\partial}{\partial x_i}\ , \qquad  \sum_{i=0}^{n-1}\, (n-i)\, y_{i+1} \, \frac{\partial}{\partial x_i}\ ,  \qquad \sum_{i=1}^{n}\,  i \, y_{i-1} \, \frac{\partial}{\partial x_i}\ , \qquad \sum_{i=1}^{n}\,  i \, y_{i}\,  \frac{\partial}{\partial x_i}\ ;
in particular, $\Der_k(R)/R\varepsilon$ is minimally generated by three homogeneous element of degree zero.
\item $L^*\cong (y_0,y_1, y_2)(1)\, .$
\item The natural map  $\Der_k(R)\to L^*$ is surjective.
\end{enumerate}
\end{proposition}
\begin{proof} We consider the polynomial ring $T=k[u,v]$, where the variables $u$ and $v$ are given degree $\frac{1}{n}$. By mapping $y_i$ to $u^{n-i}v^i$, one identifies $R$ with the Veronese subring $k[\{u^{n-i}v^{i}\}]=\oplus_{j \in {\mathbb Z}_{\ge0}} T_j$ of $T$.
By \autoref{corscrolls} the $R$-submodule $\, \Der_k(R)$ of $\Der_k(T)$ is minimally generated by the elements of degree zero $u\frac{\partial}{\partial u}$, $v\frac{\partial}{\partial u}$,  $u\frac{\partial}{\partial v}$, $v\frac{\partial}{\partial v}$.

Consider the natural map  $S \to T$ of $k$-algebras with $x_i \mapsto u^{n-i}v^i$. It induces a $T$-linear map $\Omega_k(S)\otimes T \to \Omega_k(T)$ with $dx_i \mapsto (n-i)u^{n-i-1}v^idu + i u^{n-i}v^{i-1} dv$. Dualizing into $T$, we obtain a map $\, {\rm Der}_k(T) \to {\rm Der}_k(S,T)\, $ with $\, \frac{\partial}{\partial u}\mapsto \sum_{i=0}^{n-1}\, (n-i)\, u^{n-i-1}v^i  \, \frac{\partial}{\partial x_i}\, $ and $\, \frac{\partial}{\partial v}\mapsto \sum_{i=1}^{n}\, i\, u^{n-i}v^{i-1}  \, \frac{\partial}{\partial x_i}$.

Using the identification of ${\rm Der}_k(R)$ as an $R$-submodule of ${\rm Der}_k(T) $ and ${\rm Der}_k(S,T),$

\begin{tikzcd}[row sep=small]
{\rm Der}_k(T)  \arrow{rr}  &  & {\rm Der}_k(S,T)\\
& \arrow[ul, phantom, sloped, "\supset"]~{\rm Der}_k(R)  \arrow[ur, phantom, sloped, "\subset"]&
\end{tikzcd}
\noindent
the generators $$u\frac{\partial}{\partial u}\ ,  \qquad  v\frac{\partial}{\partial u}\ ,  \qquad   u\frac{\partial}{\partial v}\ ,  \qquad  v\frac{\partial}{\partial v}$$ of ${\rm Der}_k(R)$ become $$ \sum_{i=0}^{n-1}\, (n-i)\, y_i \, \frac{\partial}{\partial x_i}\ , \qquad  \sum_{i=0}^{n-1}\, (n-i)\, y_{i+1} \, \frac{\partial}{\partial x_i}\ ,  \qquad \sum_{i=1}^{n}\,  i \, y_{i-1} \, \frac{\partial}{\partial x_i}\ , \qquad \sum_{i=1}^{n}\,  i \, y_{i}\,  \frac{\partial}{\partial x_i}\ .

We now prove (b). We may assume that $k$ is perfect. As the rational normal curve is smooth, \autoref{1.2} and
\autoref{3iso}(b)  imply that $L^*\cong \omega_R^*.$

Since $R$ is a determinantal ring, one has $\omega_R \cong (y_0,y_1)^{n-2}(n-3).$
It follows that
\begin{eqnarray*}
\omega_R^*(-1) & \cong & y_0^{n-2}R:_R(y_0,y_1)^{n-2}R \\
&=& (u^{n(n-2)}T\cap R):_R  (u^n,u^{n-1}v)^{n-2} R\\
&=& (u^{n(n-2)}T:_T  (u^n,u^{n-1}v)^{n-2} R) \cap R\\
&=& (u^{n(n-2)}T:_T  (u^n,u^{n-1}v)^{n-2} T) \cap R\\
&=& (u^{n-2}T) \cap R \\
&=& (y_0, y_1, y_2).
\end{eqnarray*}

To prove part (c) recall that according to \autoref{1.2}, the natural map $\, \Der_k(R) \lto L^*$ induces an embedding $\, \Der_k(R)/R\varepsilon \hookrightarrow L^*.$ Now use that $\, \Der_k(R)/R\varepsilon \,$ is minimally generated by $3$ homogenous elements of degree zero according to (a) and $L^*$ is generated by $3$ homogeneous elements of degree zero by (b). \end{proof}

\noindent{\bf Acknolwedgment.} Part of this paper was written at
Mathematisches Forschungsinstitut Oberwolfach (MFO), Germany,
while the third and fifth authors participated in a Research in Pairs. Part of this paper was written also at the  Instituto de Matem\'atica Pura e Aplicada (IMPA) Rio de Janeiro, Brazil. The authors are very appreciative of the hospitality offered by MFO and IMPA.

% \bib, bibdiv, biblist are defined by the amsrefs package.
\begin{bibdiv}
\begin{biblist}

\bib{A}{book}{
author={Ang\'{e}niol, Bernard},
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\end{document}

$\% Hamid:\{$  A historical  Note. During the 1880s and 1890s, there were multiple competitions aimed at determining the general solutions of ordinary differential equations with polynomial coefficients. Two renowned mathematicians, M. Darboux and M. Painlev\'e, emerged as winners of those  prizes. The intricacies of the problem attracted the attention of H. Poincar\'e for a decade.  It seems that the most interesting question for him was to find a criterion to decide when a Differential Equation of (Degree 1 and Order 1) has an algebraic integral.  An immediate idea to find such an integral, as stated in the first paragraph of \cite{P}, is
\begin{quote}
To bound the degree of integrals in terms of the degree of the coefficients.
\end{quote}
However, Poincar\'e reveals in the second page of \cite{P} that, in general, no such bound exists unless certain conditions are imposed on the integrals, such as irreducibility. Despite this, he persists in seeking  bounds in specific cases.

The whole paper \cite{P} is restricted to a  special ODE which has already been solved by Darboux. Poincar\'e establishes a connection between the degree of integral curves and what he calls 'the Dimension,' referring to the degree of coefficients.  In the last paragraph of the paper he states that: If that special  ODE has an irreducible  integral of genus 1 or 2 then the degree of the integral curve is bounded in terms of the  Dimension.

Poincar\'e did not consider the results of  his paper of great value (as stated in the second-to-last paragraph of the second page) due to their lack of generality. Despite this, he remains hopeful that his paper might eventually lead to the development of criteria for verifying the non-integrability of an equation in the future.$\}$
\vspace{0.5cm}

This paper is concerned with the structure of the module of derivations and its interplay with singularities and vector fields of varieties.
Modules of derivations are not well understood -- despite great advances on the Zariski-Lipman conjecture \cites{Ho, SvS, Flen, Kall} for instance,  there is still no complete characterization for when they are free.
The paper focuses on Poincar\'{e}'s problem on the degrees of vector fields.

In 1891, Poincar\'{e} asked the following question that became known as Poincar{\'e}'s problem \cite{P}:
How can one decide whether a homogeneous differential equation is given by a polynomial vector field
$\mathcal F$
$\mathbb P^2_{\mathbb C}$
has a rational solution? This question has been rephrased as the problem to find upper bounds for the degree of any curve $X$ to which $\mathcal F$ is tangent, possibly in terms of the degree of $\mathcal F$.

According to \cite{EK2003}*{p.57}, `This question is fundamental but difficult, and it has stimulated a lot of research for well over a century' (see \cites{CL, C, CC, S1, dPW, CCF, S2, E, LN, Per, EK, EK2002, EK2003, CE, GM}). It has often been addressed
in greater generality, for curves and even varieties in ${\mathbb P}_{\mathbb C}^n$,
and invariants other than the degrees of the variety and the vector field
have been considered, because
even for plane curves
bounds only involving degrees are not always possible \cites{C,CC,LN}.

An algebraic  vector field on ${\mathbb P}_k^n$
is said to {\it leave} a subvariety $V
\subset {\mathbb P}_k^n$ {\it invariant} if it defines a tangent direction to $V$ at each point in a dense open subset.
With the goal in mind of finding curves left invariant by a vector field on ${\mathbb P}_{\mathbb C}^2$,
in 1891 Poincar\'{e} asked whether
the degree of such curves can be bounded above in terms of the degree of the vector field \cite{P}. This would limit the search for such curves
to a finite dimensional vector space of polynomials. Unfortunately, as observed by Carnicer \cite{C},
Poincar\'{e}'s question has a negative answer, but nevertheless the broader goal remains to relate degrees of
vector fields to invariants of curves, or varieties, that they leave invariant.
According to \cite{EK2003}*{p. 57} `This question is fundamental, but difficult, and it has stimulated a lot of research for well over a century'. In the last 30 years, it has been investigated
by many mathematicians mainly from the point of view of several complex variables and analytic spaces, dynamical systems and ergodic theory, algebraic geometry, and differential geometry (see \cites{C, CL, S1, dPW, CCF, S2, E, L, Per, EK, EK2002, EK2003, CE, GM}).

Poincar{\'e} asked
the following question that became known as Poincar{\'e}'s problem [P]. How can one decide whether a polynomial
vector field $\mathcal F$
on the complex projective plane is algebraically integrable,
equivalently, whether a singular algebraic foliation of the plane has a leaf whose Zariski closure is an
algebraic curve $X$? He also observed that an answer could be found by explicit computation if
one had an apriori upper bound for the degree of the curve, possibly in terms of the degree
of the vector field. Thus the broader problem arose to find upper bounds for the degree of $X$ in terms of $\mathcal F$ or,
from the opposite perspective, to find lower bounds for the degree of $\mathcal F$ in term of $X$.
This problem has generated considerable
interest; see for instance \cites{CL, C, CC, S1, dPW, CCF, S2,  E, L, Pe, EK, EK2002, EK2003, CE, GM}.
It has often been addressed
in greater generality, for curves and even varieties in ${\mathbb P}_{\mathbb C}^n$,
and invariants other than the degrees of the variety and the vector field
have been considered, because
even for plane curves
bounds only involving degrees are not always possible (\cites{C, CC, LN}).

In this article, we study this problem from
the opposite perspective, by establishing
{\it lower} bounds on the degree of the vector field in terms of invariants of the variety $X$. This approach
has the advantage that all the vector fields on ${\mathbb P}^n_k$ tangent to $X$ can be encoded in a
single module,
\vspace{0.19cm}

\centerline{$\Der_k(R)/\m^{-1}\varepsilon\, .$}

\vspace{0.18cm}
\noindent
Here $R$ is the homogeneous coordinate ring of a reduced and equidimensional
subscheme $X \subset \mathbb P^n_k $ over an algebraically closed field of characteristic zero, $\m$ denotes its homogeneous maximal ideal, and $\varepsilon$ its Euler derivation. If $\depth \, R \geq 2$, then $\m^{-1}\varepsilon =R\varepsilon$
and the module above is essentially equivalent to the module of
derivations, $\Der_k(R)$. Now, our reformulation of Poincar\'{e}'s problem
becomes:{\it  Find lower bounds for the initial degree of $\, \Der_k(R)/\m^{-1}\varepsilon$.

In collaboration with Marc Chardin, Hamid Hassanzadeh, Aron Simis, and Bernd Ulrich, we plan to investigate this question using tools from commutative algebra. The most efficient algebraic reformulation of the problem
arises from the opposite perspective, when we ask for
{\it lower} bounds on the degree of the vector field in terms of invariants of the variety. Let $k$ be an algebraically closed field of characteristic zero and let $V$ be a subvariety of ${\mathbb P}_k^n$ with homogenous coordinate ring $R$ (most of these assumptions are not necessary, but will help make the description of the project less technical). Let $\m$ be the homogenous maximal ideal  of $R$, $\Der_k(R)$ the module of $k$-derivations of $R$, and $\varepsilon\in \Der_k(R)$ the Euler derivation. We can encode all the vector fields on ${\mathbb P}^n_k$ that leave $V$  invariant in the single
$$\Der_k(R)/\m^{-1}\varepsilon\, .$$
Thus a reformulation of Poincar\'{e}'s problem is to find lower bounds for the initial degree of $\, \Der_k(R)/\m^{-1}\varepsilon$.

My work in this area is joint with Marc Chardin, Claudia Polini, Aron Simis and, most recently, Hamid Hassanzadeh.
To introduce the algebraic setting, let $k$ be an algebraically closed field of characteristic zero and $X$ a reduced and equidimensional subscheme of
$\mathbb P_k^n$, with homogeneous coordinate ring $R$. Let $\mathfrak m$ denote the homogeneous maximal ideal and
$\varepsilon$ the Euler derivation of $R$. The natural embedding ${\mathfrak m}^{-1} \varepsilon \subset {\rm Der}_k(R)$
gives rise to a graded $R$-module $N:= {\rm Der}_k(R)/{\mathfrak m}^{-1}  \varepsilon $, which can also be described as ${\rm Der}_k(R)/R\varepsilon$
modulo $R$-torsion. The homogeneous elements of $N$ of degree $\alpha$, say, correspond to the vector fields $\mathcal F$ of degree $\alpha +1$ on ${\mathbb P}^n_k$
that are tangent to $X$. Thus, a version of Poincar{\'e}'s problem is: Establish lower bounds for the initial degree of the module $N$.

\noindent
We mostly treat the case when $X$ is a (usually irreducible) curve  $\mathcal C$. One of the main result of this paper is an estimate in terms of the $a$-invariant of $R$,
the number of singular points, and the Loewy multiplicity modulo the Jacobian ideal.
It says that if $\mathcal C$ has only plane singularities, then

\vspace{.26 cm}
\centerline{
\indeg (\Der_k(R)/\m^{-1}\varepsilon )\geq a(R) +1 + | {\rm Sing}(\C) | - {\rm Lmult}(R/ {\rm Jac}(R)) \, ;
\vspace{.26cm}

\noindent
the {\it Loewy multiplicity} ${\rm Lmult}$ of a module is a tighter version of the
usual multiplicity in which Loewy length replaces length in the associativity formula. If $\mathcal C$ has only ordinary nodes as singularities, then
$|{\rm Sing}(\C) | - {\rm Lmult}(R/ {\rm Jac}(R))=0$ and we obtain
the inequality $\indeg (\Der_k(R)/\m^{-1}\varepsilon )\geq a(R) +1$, which
we prove to be an equality when $\mathcal C$ is arithmetically Gorenstein.
The case of ordinary nodes  had
been treated before with the additional assumption that, first, $\C$ is a plane curve \cite{CL},
then, $\C$ is a complete intersection \cite{CCF}, and, finally, $\C$
is arithmetically Cohen-Macaulay \cites{E, EK2002}.

One of the results we have proved so far implies
\begin{equation}\label{EQ0}\indeg (\Der_k(R)/\m^{-1}\varepsilon )\geq a(R) +1\end{equation}
if $V=\mathcal C$
is a curve with only ordinary nodes as singularities;
here $a(R)=-\, \indeg \ \omega_R \, $ denotes the  $a$-invariant of $R$, which
equals ${\rm reg}  \, R -2 \, $ if $R$ is Cohen-Macaulay. The same bound  had
been established before with the additional assumption that, first, $\C$ is a plane curve \cite{CL},
then, $\C$ is a complete intersection \cite{CCF}, and, finally, $\C$
is arithmetically Cohen-Macaulay \cites{E, EK2002}.

In light of Carnicer's examples, one should
expect that estimates for more general curves require information about the singularities,
as encoded in the $a$-invariant, the Castelnuovo-Mumford regularity, or the multiplicity, modulo certain Jacobian
ideals, for instance. We propose yet another, tighter invariant, which we call {\it Loewy multiplicity}, in which
Loewy length replaces length in the associativity formula for multiplicity: the Loewy multiplicity of a Noetherian module $M$ over
a local ring $R$ is defined as
$\ell{\rm mult}(M):=\sum \ell \ell (M_{\p}) \cdot e(R/\p)$, where $\p$ ranges over the
primes of maximal dimension in ${\rm Supp}(M)$; it satisfies $\ell{\rm mult}(M) \leq e(M)$. One
of our main results so far says that
\begin{equation}\label{EQ} \indeg (\Der_k(R)/\m^{-1}\varepsilon )\geq a(R) +1 + | {\rm Sing}(\C) | - \ell{\rm mult}(R/ {\rm Jac}(R))
\end{equation}
if $\C$ is a curve with only plane singularities; here ${\rm Sing}$ denotes the singular locus and ${\rm Jac}$ the Jacobian ideal. Notice that $|{\rm Sing}(\C) | -\ell{\rm mult}(R/ {\rm Jac}(R))=0$
if and only if  $\C$ has at most ordinary nodes as singularities, hence Formula (\ref{EQ}) implies  (\ref{EQ0}). The question remains
what happens if not all singularities are planar.

We also generalized from plane curves to arbitrary curves, earlier results of
du Plessis and Wall \cite{dPW} and of Esteves and Kleiman \cite{EK} that use the (usual)
multiplicity $e(R/{\rm Jac}(R))$. Furthermore, inspired by ideas from \cite{EK2002}
we have proved new estimates for plane curves: If $\mathcal F$ is a vector field  tangent to
$\mathcal C$, the Jacobian ideal ${\rm Jac}(R)$ coincides with an ideal
$I_{\mathcal F}$ defining the singular locus of $\mathcal F$,
up to a degree shift that depends on the degree of $\mathcal F$. Thus
to bound this degree shift, and hence $\rm{deg} \, \mathcal F$, from below, it
suffices to compare the two multiplicities $e(R/{\rm Jac}(R))$ and
$e(R/I_{\mathcal F})$ and, in particular, to bound the latter from below. We
prove a non-vanishing result for maps between the local cohomology
modules of Koszul cycles
that yields lower bounds for the regularity
of the saturation $I_{\mathcal F}^{\rm sat}$ and hence for $e(R/I_{\mathcal F})$.
Thus we obtain, for a plane curve of degree $d$ and geometric genus

\vspace{.2cm}
\centerline{
$\indeg (\Der_k(R)/R\varepsilon )\geq d-\frac{3}{2} - \sqrt{2e(R/{\rm Jac}(R)) +2p_g-d^2+3d-\frac{7}{4}} \,.
\vspace{.2cm}

\noindent
This estimate is sharp  for low genus and for curves with only ordinary nodes as singularities.

So far we can prove that if $X\subset {\mathbb P}^n_k$ is a curve and $X$ has only ordinary nodes as singularities, then
the initial degree of $N$ is at least $a(R)$, the $a$-invariant of $R$. With the same assumption on the singularities, equivalent
had been proved
first for plane curves \cite{CL}, then for complete intersection curves \cite{CCF}, and finally for
arithmetically Cohen-Macaulay curves \cites{E, EK2002}. These previous results also show that ${\rm indeg} \, N \geq a(R) + 1$ if in addition $X$ is
irreducible, and we wish to recover this improved bound without the assumption of arithmetic Cohen-Macaulayness.
Moreover, we would like to strengthen our estimate, replacing the $a$-invariant by the Casteluovo-Mumford regularity minus 2,
${\rm reg} \, R -2$, and we
hope to show that the resulting inequalities are sharp or close to being sharp.
In this vein we have proved, so far,  an {\it upper} bound for the initial degree of $N$ that does not require assumptions on the singularities and that
specializes to the inequality ${\rm indeg} \, N \leq a(R)+1={\rm reg}\, R -1$
if $X$ is a complete intersection curve.

For curves that are not nodal the initial degree of $N$ can be considerably smaller, and although a great number of important results
are known this subject is still open ended. Our contribution so far is to the case of arithmetically Gorenstein curves with only complete
intersection singularities, where we prove that ${\rm indeg} \, N \geq a(\overline{R})= {\rm reg} \, {\overline R} -2$, the $a$-invariant of the normalization of $R$.
Again, we wish to improve this bound by 1 if $X$ is irreducible.

The proof of inequality (\ref{EQ}) has two main ingredients, which are of independent interest.
Inspired by the use of general projections in \cite{E},
we prove quite generally that if $A \subset R$ is a finite and birational extension of standard graded domains
over a perfect field and $A$ is Gorenstein of dimension at least two, then these upper and lower bounds hold,
$$ \indeg (\Der_k(A)/A  \varepsilon_A) +a(A)-a(R) \geq  \indeg (\Der_k(R)/\m^{-1} \varepsilon_R) \geq \indeg (\Der_k(A)/A  \varepsilon_A) -a(A)+a(R) \, .$$
In addition, for hypersurfaces of arbitrary dimension with only isolated singularities, we are able to bound $\indeg (\Der_k(R)/\m^{-1}\varepsilon)$
from below in terms of the $a$-invariant of $R/ {\rm Jac}(R)$. Applying these two results to
a curve $\C$ with only plane singularities, we prove inequality (\ref{EQ}) by general projection to a plane curve. The general projection
does not change the singularities of $\C$ and introduces only ordinary nodes as additional singularities, hence the
difference $| {\rm Sing}(\C) | - \ell{\rm mult}(R/ {\rm Jac}(R))$ remains constant.
\smallskip

Curiously, for an arithmetically Gorenstein curve with only ordinary nodes as singularities, we deduce from
the inequality
(\ref{EQ0}) that the Euler derivation does not
generate a direct summand of $\Der_k(R)$. This suggests the next question about homogeneous coordinate
rings of any subvarieties of $ {\mathbb P}_k^n$, which, like much of this section, aims at a better
understanding of modules of derivations. Little is known about these modules, besides the (not obvious) fact that they cannot be
free if $V$ is not a linear subspace \cite{Ho}.

Returning to  the problem of finding lower bounds for the initial degree of $\, \Der_k(R)/\m^{-1}\varepsilon$ for curves, let $\C$ be a curve of degree $d$. We map
a sufficiently general two-dimensional complete intersection $S$ onto $R$, and write $\delta$ for the Castelnuovo-Mumford
regularity ${\rm reg} \, S$ and $J$ for the image in $R$ of the Jacobian ideal ${\rm Jac}(S)$.
Another approach to the problem, which employs tools proper to Commutative Algebra, is via the embedding
\begin{equation}\label{EQ3} \Der_k(R)/\m^{-1}\varepsilon  \hookrightarrow J^{-1}(2-\delta) \, .\end{equation}
We relate the ideal $J$ to the singular locus of a vector field of minimal degree leaving $\C$ invariant
and use an estimate for the regularity of this singular locus from \cite{EK2003}, to prove that
\begin{equation}\label{EQ2} \indeg (\Der_k(R)/\m^{-1})\ge \delta -2 -\frac{e(R/J)-\delta}{d-1} \, .\end{equation}
This generalizes a bound that was known for plane curves
\cite{dPW}. If $\C$ is a complete intersection, then the multiplicity $e(R/J)$ is the {\it total Tjurina number} of $\C$,
the sum of the local Tjurina numbers, which give the dimension of the formal moduli space of each singularity.

The essential quantities in Formulas (\ref{EQ}) and (\ref{EQ2}) satisfy the inequality $\ell{\rm mult}(R/{\rm Jac}(R)) \leq e(R/J)$,
but the two estimates are nevertheless incomparable. Hence we propose to find a bound that is a common improvement of either formula.

\smallskip

We pause to explain what enters into
the proofs
of our two estimates, ${\rm indeg} \, N \geq a({R})$ and ${\rm indeg} \, N \geq a(\overline{R})$ for  curves that are nodal or arithmetically Gorenstein
local complete intersections, respectively. We first show that for any curve, to estimate ${\rm indeg} \, N$ from below
it suffices to find upper bounds for
the Castelnuovo-Mumford regularity or even the $a$-invariant of a partial Jacobian ideal times a canonical ideal of $R$.
By a mixture of global and local considerations, we are able to prove such bounds, essentially, by relating the partial Jacobian
ideal to the conductor, whose regularity can be estimated more easily (see, for instance, \cite{EU}). More specifically,
we use linkage to replace the partial Jacobian ideal by the full Jacobian ideal $\mathcal{J}(R)$, and
then exploit the local information about the singularities to prove a relationship between $\mathcal{J}(R)$ and the conductor $\mathfrak{c}(R)$
that is considerably stronger than the well known inclusion $\mathcal{J}(R) \subset \mathfrak{c}(R)$; in the arithmetically
Gorenstein case, we also compute the initial degree of the conductor from a known equality involving the Dedekind
complementary modules (canonical ideals) of $R$ and $\overline R$.

If $X$ is an arbitrary curve, Poincar{\'e}'s question can be interpreted more broadly as the problem to relate the degree, or other
of the vector field to invariants of the curve. Invariants that have been considered include the regularity and multiplicity of $R/{\mathcal J(R)}$,
and the multiplicity of ${\overline R}/R$ as an $R$-module. We propose yet another invariant, defined in terms of local Loewy lengths, that we will describe
below for hypersurfaces of arbitrary dimension.

If $X$ is a hypersurface of arbitrary dimension, we allow for $X$ to have isolated singularities and we observe that, as in the case of plane curves,
bounding $\indeg N$ from below amounts to bounding $a(R/\mathcal J(R))$ from above.  We estimate this $a$-invariant
in terms of local Loewy lengths, $$a(R/\mathcal J(R)) \leq \sum \ell \ell({\mathcal O}_{X,p}/{\mathcal J}({\mathcal O}_{X,p})) -2 ,$$
where $p \in {\rm Sing}(X)$ and the Loewy length $\ell \ell(A)$ of an Artinian local ring $(A, \m)$ is the smallest integer $s\geq0$ so that
$\m^s=0$. The Loewy length of the ring ${\mathcal O}_{X,p}/{\mathcal J}({\mathcal O}_{X,p})$ is at most its length,
and for an isolated hypersurface singularity this length is
the Tjurina number,
the embedding dimension of the formal moduli space of the singularity. The sum of the local Tjurina numbers, which is nothing but
the multiplicity of the ring $R/\mathcal J(R)$, has been used before to estimate $\indeg N$
if $X$ is a plane curve (see for instance \cites{dPW, EK, EK2003}), but
the known bounds are not comparable to our inequality above, as they use, roughly speaking, the sum of the local Tjurina numbers divided by
$a(R)+2$ in place of the sum of the local Loewy lengths. We wish to reconcile these two types of results in order
to optimize our estimate for arbitrary hypersurfaces with isolated singularities.

In order to understand how sharp the various lower bounds for $\indeg(\Der_k(R)/\m^{-1}\varepsilon)$ are, we
also plan to search for upper bounds, which is a natural goal anyway at least from an algebraic point of view.

If $R$ is the homogeneous coordinate ring of a curve, then
a presentation of the module of differentials gives
an epimorphism $\wedge^2\, \Omega_k(R) \twoheadrightarrow J(\delta-2)$ and there is a natural homomorphism
$\wedge^2\, \Omega_k(R) \to \omega_R$ called the {\it canonical class}. Dualizing both maps into $R$ yields an embedding
$\, \omega_R^{*} \hookrightarrow J^{-1}(2-\delta)$. On the other hand, if $R$ is Cohen-Macaulay, then the
cokernel of the map $\, \Der_k(R)/\m^{-1}\varepsilon  \hookrightarrow J^{-1}(2-\delta) \, $ as in (\ref{EQ3}) is
concentrated in degrees at most $a(R)$, and we deduce that
\begin{equation}\label{EQ4} \indeg(\Der_k(R)/\m^{-1}\varepsilon) \leq  {\rm max}\{\indeg \, \omega_R^{*},a(R)+1\} \end{equation}
for any arithmetically Cohen-Macaulay curve. This gives the definitive bound $\indeg(\Der_k(R)/\m^{-1}\varepsilon) \leq  a(R)+1$
in the arithmetically Gorenstein case,
which is an equality by (\ref{EQ0}) if the curve has only ordinary nodes as singularities. Using this estimate and
general projections to plane curves, we obtain upper bounds for arbitrary curves in terms of $a(R)$ and
the degree of the curve, though these bounds are not optimal yet. We also show that, in general,
bounds only in terms of the $a$-invariant cannot exist.