| 摘要: |
| 摘要：对于特征0的光滑射影代数曲面S上的半稳定丛E，Bogomolov \cite{Bog78}证明了$\Delta(E)\ge 0$。Bogomolov不等式在代数几何中有着广泛的应用。已知在正特征情形下，经典的Bogomolov不等式在某些情况下并不成立，例如在某些拟椭圆曲面上\cite{Ray78}。本文将研究拟椭圆曲面上的Bogomolov型不等式。
假设k是特征p的代数闭域，S是光滑射影曲面，f:S → B是从S到正规曲线B的拟椭圆纤维化，E是S上的半稳定凝聚层。本文建立了一个Bogomolov型不等式：
\[
\Delta(E)+\left(\frac{r(r-1)}{p}\right)^2\cdot c_0\ge0,
\]
其中r是E的秩，c_0 = c_0(a,b)是依赖于a,b的常数，a ∈ ?是使得K_S ≡ aF成立的唯一有理数，b是底曲线B的亏格。 |
| 关键词: 正特征, 拟椭圆曲面, Bogomolov不等式 |
| DOI： |
| 分类号:O187.2 |
| 基金项目: |
|
| Bogomolov Inequality on Quasi-elliptic Surfaces |
|
MengLve
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| Abstract: |
| Abstract: For a semistable vector bundle E on a smooth projective algebraic surface S over characteristic 0, Bogomolov [Bog78] proved that Δ(E) ≥ 0. The Bogomolov inequality has wide applications in algebraic geometry. It is known that in positive characteristic, the classical Bogomolov inequality fails to hold in certain cases, such as on some quasi-elliptic surfaces [Ray78]. This paper investigates Bogomolov-type inequalities on quasi-elliptic surfaces.
Let k be an algebraically closed field of characteristic p, S a smooth projective surface, f:S → B a quasi-elliptic fibration from S to a normal curve B, and E a semistable coherent sheaf on S. We establish the following Bogomolov-type inequality:
\[
\Delta(E) + \left(\frac{r(r-1)}{p}\right)^2 \cdot c_0 \geq 0,
\]
where r is the rank of E, c? = c?(a,b) is a constant depending on a and b, a ∈ ? is the unique rational number satisfying K_S ≡ aF, and b is the genus of the base curve B. |
| Key words: positive characteristic,quasi-elliptic surface,Bogomolov inequality |
