Effective Matsusaka’s theorem for surfaces in characteristic \(p\). (English) Zbl 1386.14133

Di Cerbo, Gabriele; Fanelli, Andrea

Algebra Number Theory 9, No. 6, 1453-1475 (2015).

A celebrated Matsusaka-Kollar theorem states that, for a smooth complex projective variety \(X\) with \(n=\dim X\) and an ample divisor \(D\) on \(X\), there exists a positive integer \(M\) depending only on the intersection numbers \((D^n)\) and \((K_X \cdot D^{n-1})\) such that \(\ell D\) is very ample for every \(\ell\geq M\).
In the paper under review, the authors consider smooth projective surfaces \(X\) over an algebraically closed field \(k\) of positive characteristic and show the following (Theorem 1.2): For an ample divisor \(D\) and a nef divisor \(B\) on \(X\), \(m D - B\) is very ample for every \(m>M\) where \(M\) is as follows: \[ M = \frac{2D. (H+B)}{D^2}((K_X + 2D)\cdot D + 1) \] where \[ H= K_X + 4D\text{ if }X\text{ is neither quasi-elliptic with }\kappa(X)=1\text{ nor of general type} \]
\[ H= K_X +8D\text{ if }X\text{ is quasi-elliptic with }\kappa(X)=1\text{ and }p=3 \]
\[ H=K_X+19D\text{ if }X\text{ is quasi-elliptic with }\kappa(X)=1\text{ and }p=2 \]
\[ H=2K_X + 4D\text{ if }X\text{ is of general type and }p\geq 3 \]
\[ H=2K_X + 19D\text{ if }X\text{ is of general type and }p=2. \] The proof uses a positive characteristic version of Fujita conjecture for surfaces, which claims lower bound of \(\ell\in{\mathbb N}\) such that \(K_X + \ell D\) is very ample. This version of Fujita conjecture has been proved by N. I. Shepherd-Barron [Invent. Math. 106, No. 2, 243–262 (1991; Zbl 0769.14006)] and H. Terakawa [Pac. J. Math. 187, No. 1, 187–199 (1999; Zbl 0967.14008)] except in the cases of quasi-elliptic and of general type. Thus, the authors focus on these exceptional cases and gives a partial answer to Fujita conjecture using the well-known relation between unstableness of rank two vector bundles and counter-examples of vanishing of the first cohomologies, e.g. Kodaira vanishing, originated from H. Tango [Nagoya Math. J. 48, 73–89 (1972; Zbl 0239.14007)] together with the bend-and-break technique (Theorem 1.4).
Using this proof technique, the authors also give an effective version of Kawamata-Viehweg vanishing theorem for surfaces in positive characteristic (Corollary 5.9). Namely, for a nef and big Cartier divisor \(D\) on \(X\), we have \(H^1(X, \mathcal{O}_X(K_X + mD))=0\) for all \(m>m_0\) where \(m_0 = \displaystyle{\frac{3}{p-1}}\) if \(X\) is quasi-elliptic with \(\kappa(X)=1\) and \(m_0= \displaystyle{\frac{2 \text{vol}(X) + 9}{p-1}}\) if \(X\) is of general type, where \(\text{vol}(X):=\displaystyle{\limsup_{n\to\infty} \frac{2 h^0(X, \mathcal{O}_X(n K_X))}{n^2}}\).

Reviewer: Yukihide Takayama (Shiga)

MSC:

14J25

Special surfaces

Keywords:

effective Matsusaka’s theorem; surfaces in positive characteristic; Fujita’s conjectures; Bogomolov’s stability; Reider’s theorem; bend-and-break; effective Kawamata-Viehweg vanisihng

Citations:

Zbl 0769.14006; Zbl 0967.14008; Zbl 0239.14007