In the paper under review, the author presents an explicit description of the rational cohomology ring \(H^0(M_4,\mathbb{Q})\) of the coarse moduli space \(M_4\) of non-singular complex curves of genus four. By earlier results of J. Harer and D. Zagier [Invent. Math. 85, 457–485 (1986; Zbl 0616.14017)], the Euler characteristic of \(M_4\) was previously known to be equal to 2, and some fundamental algebraic cohomology classes in the Chow ring of \(M_4\) had been computed by C. Faber [Ann. Math. (2) 132, No. 3, 421–449 (1990; Zbl 0735.14021)] in a more general context.These partial results on the ratinal cohomology of \(M_4\) are now completed by the author’s present main results culminating in a precise calculation of the Poincaré-Serre polynomial of the moduli space \(M_4\).
The basic strategy for investigating the cohomology of \(M_4\) is to stratify this space in the form \(C_2\subset\overline C_1\subset\overline C_0= M_4\), where \(C_2\) denotes the locus of hyperelliptic curves, \(C_1\) the locus of curves whose canonical model is the complete intersection of a cubic surface and a quadric cone in \(\mathbb{P}^3\), and \(C_0\) stands for the locus of curves whose canonical model is the complete intersection of a cubic surface and a smooth quadric surface in \(\mathbb{P}^3\). Then the rational cohomology of these strata is tackled separately. As for the crucial loci \(C_0\) and \(C_1\), the author’s approach is based on suitable modifications of recent methods developed C. Peters and J. Steenbrink [Mosc. Math. J. 3, No. 3, 1085–1095 (2003; Zbl 1049.14035)], on the one hand, and by V. A. Vassiliev [Proc. Steklov Inst. Math. 225, 121–140 (1999); translation from Tr. Mat. Inst. Steklova 225, 132–152 (1999; Zbl 0981.55008)] on the other, which are comprehensively elaborated in the first part of the paper. The application of these refined topological tools to the rational cohomology of the spaces \(C_0\) and \(C_1\) leads then to the following two theorems stated in terms of Poincaré-Serre polynomials:
(1) The space \(C_0\) has Poincaré-Serre polynomial \(1+ t^5u^6\).
(2) The space \(C_1\) has the rational cohomology of a single point.
Together with the fact that the hyperelliptic locus \(H_g\) has always the rational cohomology of a point, a classical result that is reproved here in the spirit of the current paper, the specific result, (1) and (2) yield the author’s main theorem stating that the Poincaré-Serre polynomial of the whole moduli space \(M_4\) is precisely the polynomial \(1+ t^2u^2+ t^4u^4+ t^5u^6\).
In a subsequent paper [Math. Ann. 338, No. 1, 207–239 (2007; Zbl 1126.14030)], J. Bergström and the author have extended the approach presented here to study pore closely the rational cohomology of the Deligne-Mumford compactification \(\overline M_4\) of \(M_4\), that is, the rational cohomology of the moduli space of stable curves of genus 4. Further applications of the Vassiliev-Gorinov method to computing the rational cohomology of other moduli spaces of low-genus curves can be found in the author’s additional paper “Rational cohomology of \(M_{3,2}\)” [Compos. Math. 143, No. 4, 986–1002 (2007; Zbl 1126.14033)].