Local positivity of ample line bundles.

*(English)*Zbl 0866.14004Let \(L\) 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 \(t\).

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 |