Surfaces with \(p_g=q=3\) (Q2782656)

The smallest possible value of the Euler-Poincaré characteristic \(\chi({\mathcal O}_S) \) for a smooth complex surface \(S\) of general type is \(1\). If \(\chi({\mathcal O}_S)=1\) and \(S\) is minimal, then \(1\leq K^2_S\leq 9\). Furthermore, if \(S\) has \(q>0\), the inequality \(K^2_S\geq 2p_g(S)\) [\textit{O. Debarre}, Bull. Soc. Math. Fr. 110, No.~3, 319-342 (1982; Zbl 0543.14026)] implies \(p_g\leq 4\), and if \(p_g=4\), \(S\) is the product of two curves of genus \(2\) [\textit{A. Beauville}, Appendix to the paper by O. Debarre (loc. cit.)].NEWLINENEWLINENEWLINEIn the paper under review the minimal complex surfaces of general type with \(p_g=q=3\) are completely classified. Namely it is shown that such a surface is either the symmetric product of a curve of genus \(3\) or a free \({\mathbb{Z}}_2\)-quotient of the product of a curve of genus \(2\) and a curve of genus \(3\). Furthermore a description of the moduli space of surfaces of general type with \(p_g=q=3\) is given and the degree of the bicanonical map is studied. These two types of surfaces had already been described by \textit{F. Catanese, C. Ciliberto} and \textit{M. Mendes Lopes}, Trans. Am. Math. Soc. 350, No.~1, 275-308 (1998; Zbl 0889.14019), who also showed that the first type is the only example with \(K^2_X=6\) and the second type is the only example with a pencil of curves of genus 2.NEWLINENEWLINENEWLINEThe present paper is interesting not only because of the complete classification presented but also because of the innovative approach used, which is completely independent of the methods used in the above cited paper. The main result is obtained by a beautiful use of the generic vanishing theorems of Green and Lazarsfeld and Fourier-Mukai transforms, and it is probably one of the first instances of the use of such tools in the study of surfaces of general type. It should be pointed out that almost contemporarily \textit{G. P. Pirola} [Manuscr. Math. 108, No.~2, 163-170 (2002; Zbl 0997.14009)], using methods similar to the ones of Catanese et alii (loc. cit.) also obtained the main classification result.