具有大量互不相交(-2)-曲线的数值Godeaux曲面及其应用
Numerical Godeaux Surfaces with many disjoint $(-2)$-curves and Applications
AI总结：
本文证明数值Godeaux曲面的互不相交(-2)-曲线至多6条且界最优，将该结果应用于细化两类p_g=0的光滑极小一般型曲面对合的分类。
AI中文摘要：
本文在复数域上证明，数值Godeaux曲面至多包含6条互不相交的(-2)-曲线，且该界是最优的。作为应用，我们细化了对具有p_g=0、K²=7的光滑极小一般型曲面上对合的分类：除子的固定部分R满足R²=-1，该对合在H*(S,ℚ)上平凡作用；若商空间的极小解是一般型的，则其为包含5条互不相交(-2)-曲线的数值Campedelli曲面。另一应用涉及具有p_g=0、K²=8的光滑极小一般型曲面上的可交换对合。
英文摘要：
In this paper, over the field of complex numbers, we prove that a numerical Godeaux surface contains at most six pairwise disjoint $(-2)$-curves, and that this bound is sharp. As an application, we refine the classification of involutions on smooth minimal surfaces of general type with $p_g=0$ and $K^2=7$: the divisorial fixed part $R$ satisfies $R^2=-1$, the involution acts trivially on $H^*(S,\mathbb{Q})$, and, if the minimal resolution of the quotient is of general type, it is a numerical Campedelli surface containing five pairwise disjoint $(-2)$-curves. Another application concerns commuting involutions on smooth minimal surfaces of general type with $p_g=0$ and $K^2=8$.
