# number of generators of vanishingideal of one point

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 $$n. Now we have to show: Let $$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 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.

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

• Hi, thank you for your answer. But I still struggle with the notation. For the first map $W\otimes_k\mathcal{O}_X\rightarrow L$, is this a map of sheaves and we should see $W$ as a constant sheaf and $L=\mathcal{O}_X(L)$? And in your definition of $V$, what do you mean by $id_x$ is it the multiplication of the evaluation of a section at the point $x$, or is it the identity of the set $\{x\}$? Thank you for further help. Nov 8, 2021 at 12:13
• Sorry for your confusion. (1) I identify a Cartier divisor with a line bundle. So $L$ means $\mathcal{O}_X(L)$, as you said. (2) I identify a linear system of $L$ with a linear subspace of $H^0(X,L)$. So $W\otimes_k \mathcal{O}_X$ means a pull-back of the coherent sheaf $W$ on $\mathrm{Spec}(k)$ via $X \to \mathrm{Spec}(k)$ (note that a $k$-linear space is naturally identified with a quasi-coherent sheaf on $\mathrm{Spec}(k)$). Nov 8, 2021 at 12:38
• (3) $p\otimes \mathrm{id}_x$ means a pull-back of $p:H^0(X,L)\otimes_k \mathcal{O}_X\to L$ via the morphism $\mathrm{Spec}(k)\to X$ corresponding to the closed point $x\in X$ (note that if $k=\bar{k}$, giving a closed point on $X$ is equivalent to giving a morphism $\mathrm{Spec}(k) \to X$). Nov 8, 2021 at 12:45

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 $$s_0,\ldots,s_n$$.