zbMATH — the first resource for mathematics

Local positivity of ample line bundles. (English) Zbl 0866.14004
Let $$L$$ be a nef line bundle on an irreducible complex projective variety $$X$$. For $$x\in X$$, the Seshadri constant $$\varepsilon (L,x)$$ is defined as follows: Let $$f:Y\to X$$ be the blowing-up at $$x$$, and let $$E$$ be the exceptional divisor over $$x$$. Then $$\varepsilon (L,x): =\text{Sup} \{\varepsilon\geq 0 |f^* L-\varepsilon E$$ is nef as an $$\mathbb{R}$$-divisor on $$Y \}$$.
Main result: Let $${\mathcal B} \subset X$$ be a countable union of proper closed subvarieties and let $$\alpha$$ be a positive number such that $$L^r \cdot Y\geq(r\alpha)^r$$ for any subvariety $$Y$$ of $$X$$ with $$Y\not \subset {\mathcal B}$$, $$r=\dim Y>0$$. Then $$\varepsilon (L,x)\geq \alpha$$ for any very general point $$x$$, i.e., for any $$x$$ outside a countable union of proper closed subsets. In particular, $$\varepsilon(L,x) \geq {1\over n}\quad (n= \dim X)$$ for any very general $$x$$ if $$L$$ is nef and big.
Moreover one gets a few corollaries on adjoint series:
(1) Let $$s\geq 0$$ be an integer such that $$L^r\geq (r(n+s))^r$$ for any subvariety $$Y$$ of $$X$$ with $$Y\not \subset {\mathcal B}$$, $$r=\dim Y>0$$. Then $$|K_X+ L|$$ generates $$s$$-jets at a general point $$x$$ on $$X$$.
(2) Suppose $$L^n>0$$ and $$X$$ is smooth. Then for any $$m\geq 2n^2$$, the rational map defined by $$|K_X+ mL|$$ is birational onto its image.
The proof of the main result is based on an argument used to prove boundedness of Fano manifolds of Picard number one and the following lemma:
Let $$\{Z_t \subset V_t\}_{t\in T}$$ be a family of subvarieties of a smooth variety $$X$$ parametrized by a smooth affine variety $$T$$. Suppose that there is a family $$\{E_t\}_{t\in T}$$ of divisors in a fixed linear series $$|L|$$ on $$X$$ with $$a= \text{mult}_{Z_t} (E_t)$$ and $$b= \text{mult}_{V_t} (E_t)$$ for general $$t\in T$$. Then there is another family $$\{D_t\}_{t\in T} \in|L|$$ such that $$V_t\not \subset D_t$$ and $$\text{mult}_{Z_t} (D_t)\geq a-b$$ for general $$t$$.
Reviewer: T.Fujita (Tokyo)

MSC:
 14C20 Divisors, linear systems, invertible sheaves
Full Text: