number of generators of vanishingideal of one point

Question

I try to understand a remark in Ambro and Ito's paper successive minima of line bundles.

Let $X$ be a proper variety and $L$ a Cartier-divisor on $X$. A basis $s_0,\ldots,s_N$ of $\Gamma(L)=H^0(X,\mathcal{O}_X(L))$ gives a rational map $\Phi:X\rightarrow \mathbb{P}^N=|L|$, that is regular on $U=X\setminus Bs|L|$. They say $Bs|\mathcal{I}_x(L)|\cap U=\Phi^{-1}\Phi(x)\cap U$ for all $x\in U$.

I could show $Bs|\mathcal{I}_x(L)|\cap U\supset\Phi^{-1}\Phi(x)\cap U$.

What I tried for the other direction: Take a basis $s_0,\ldots, s_n$ of $\mathcal{I}_x(L)$ and complete it to a basis $s_o,\ldots,s_N$ of $\Gamma(L)$. Then $s_0(x)=\cdots =s_n(x)=0$ and $s_i(x)\neq 0$ for $n<i\leq N$. Now we have to show: Let $a\in Bs|\mathcal{I}_x(L)|\cap U$ then $s_0(a)=\ldots =s_n(a)=0$ (this is true by definition) and $s_i(a)/s_i(x)=\lambda$ for $n<i\leq N$ and $\lambda\in k^*$ (assume $k=\mathbb{C}$). Since I didn't see this, I tried examples, but I only had examples with $n=N-1$. So I come to my question: Is it true the the number of generators of the ring of global sections of the vanishing ideal of a point associated to a divisor is one smaller then the number of generators of the ring of global sections of the sheaf associated to a divisor?

Of course I am thankful for answers to my original question, too.

Answer 1

Assume $\mathcal{I}_x(L)$ is generated by $s_0,\ldots,s_n$ and there are two sections $s_{n+1}$, $s_{n+2}\in H^0(X,\mathcal{O}_X(L))$, which are linearly independent and do not vanish at $x$. Then the section $s_{n+1}-\frac{s_{n+1}(x)}{s_{n+2}(x)}s_{n+2}$ vanishes in $x$ and is therefore an element of $\mathcal{I}_x(L)$. This is a contradiction to the linear independence of the sections $s_0,\ldots,s_{n+2}$, so there have to be less than two sections in $H^0(X,\mathcal{O}_X(L))$ not vanishing at $x$ and linearly independent to $s_0,\ldots,s_n$.

Answer 2

First, one can check easily that the base locus of the linear system $W \subset H^0(X,L)$ is just the support of the cokernel of the natural morphism $W\otimes_k \mathcal{O}_X \to L$, i.e., $\mathrm{Bs}(W) = \mathrm{Supp}(\mathrm{coker}(W\otimes \mathcal{O}_X \to L))$.

Write $p:H^0(X,L)\otimes\mathcal{O}_X\to L$ for the natural morphism and $V := \ker(p\otimes \mathrm{id}_x : H^0(X,L)\to L\otimes k(x))$. Then $V$ may be regarded as the linear system $\mathcal{I}_x(L)$, and $V \neq H^0(X,L)$ (because $x\in U$). Moreover, for any point $q\in \mathbb{P}(H^0(X,L))$, $q = \Phi(x)$ if and only if the kernel of the corresponding linear map $q:H^0(X,L) \to k$ is $V$. Hence for any closed point $x'\in U\subset X$, the followings are equivalent:

• $\Phi(x') = \Phi(x)$
• $V = \ker(p\otimes \mathrm{id}_{x'})$
• $x'\in \mathrm{Supp}(\mathrm{coker}(V\otimes \mathcal{O}_X \to L))$.

This implies that $\Phi^{-1}(\Phi(x))\cap U = \mathrm{Supp}(\mathrm{coker}(V\otimes \mathcal{O}_X\to L)) \cap U = \mathrm{Bs}(|\mathcal{I}_x(L)|)\cap U$.