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arXiv 1202.26562012-02math.AG

Severi不等式的一种轨道方法

An orbifold approach to Severi Inequality

Lei Zhang

备注：: This paper has been withdrawn, because Severi problem has been solved completely by Rita Pardini,Xin Lv et

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摘要：

对于具有$Albdim(S) = 2$的一般型$S$的光滑极小曲面，Severi不等式表示$K_S^2 \geq 4χ(S)$，Pardini证明了这一点.当等式成立时，期望$S$是一个双有理的复盖，复盖在一个阿贝尔曲面上，沿着一个至多具有可忽略奇点的因子分支。Manetti在$K_S$充足时证明了这一点。本文将Manetti方法应用到$S$的正则模型上，在一些附加的假设条件下证明了Severi不等式，并用$K_S^2 = 4χ(S)$刻画了曲面，并利用分歧因子给出了Abel曲面上双覆盖的一个刻画.

英文摘要：

For a smooth minimal surface of general type $S$ with $Albdim(S) = 2$, Severi inequality says that $K_S^2 \geq 4χ(S)$, which was proved by Pardini. It is expected that when the equality is attained, $S$ is birational to a double cover over an Abelian surface branched along a divisor having at most negligible singularities. This was proved when $K_S$ is ample by Manetti. In this paper, we applied Manetti's method to the canonical model of $S$, with some additional assumptions we proved Severi inequality and characterized the surfaces with $K_S^2 = 4χ(S)$.In addition, we gave a characterization of the double cover over an Abelian surface via the ramification divisor.

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