Masaaki Murakami (Kagoshima University)

Surfaces with c_1^2=9 and \chi=5 whose canonical classes are divisible by 3

Abstract

Study of surfaces with $p_g =4$ has a long history, even from the era of classical Italian school. In this talk, I shall explain my study on surfaces with $c^2 = 9$ and $\chi=5$ whose canonical classes are divisible by $3$ in the integral cohomology groups, where $c_1^2$ and $\chi$ denote the first Chern number of a surface and the Euler characteristic of the structure sheaf, respectively. The main results are: any such surface $S$ is essentially a $(6, 10)$--complete intersection in the weighted projective space $\mathbb{P} (1,2,2,3,5)$; its canonical map $\varPhi_{|K|} : S - - \to \mathbb{P}^3$ is either a birational map onto a singular sextic (general case) or a generically two--to--one mapping onto a cubic surface (special case); the coarse moduli space $\mathcal{M}$ of such surfaces is a unirational variety of dimension $34$. As a byproduct, we can rule out a certain case (Case (ii), Proposition 1.7) mentioned in Ciliberto--Francia--Mendes Lopes [1].

[1] Ciliberto, C., Francia, P., Mendes Lopes, M., Remarks on the bicaninical map for surfaces of general type, Math. Z. 224 (1997), 137--166.