Z_2-商的秩为r的超椭圆纤维化曲面的斜率

The Slopes of Hyperelliptic Surfaces with Z_2-Quotient Rankr

设Ｓ是一般型的相对极小曲面，ｆ：Ｓ→Ｃ是亏格ｇ的超椭圆纤维化．本文中我们证明了如果 Ｓ的代数基本群的垂直部分的极大挠 ２商为，那么其斜率且等号成立仅当 Ｓ上的超椭圆对合所诱导的二次复盖的分歧除子 Ｒ仅有（ｒ＋１→，＋１）（当ｒ为偶数）型奇点，或（ｒ＋２→ｒ＋２）（当ｒ为奇数）型奇点．

Let S be an hyperelliptic surface of general troe, f: S → C a genus g hyperelliptic fibration. In this paper, we prove that if the mtximal Z2-quotient rank of the vertical part of the fundamental group of S is r, then its slope with equality hold only if the branch locus R of the double cover induced by the hyperelliptic involution on S has(r + 1 → r + 1) type singularities(if r is even), or (r + 2 → r + 2) type singularities(if r is odd).