Algebraic varieties with automorphism groups of maximal rank

We confirm, to some extent, the belief that a projective variety X has the largest number (relative to the dimension of X) of independent commuting automorphisms of positive entropy only when X is birational to a complex torus or a quotient of a torus. We also include an addendum to an early paper though it is not used in the present paper.Comment: Mathematische Annalen (to appear

The Kodaira dimension of the moduli of K3 surfaces

The moduli space of polarised K3 surfaces of degree 2d is a quasi-projective variety of dimension 19. For general d very little has been known about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d>61 and for d=46,50,54,58,60.Comment: 47 page

Extending Torelli map to toroidal compactifications of Siegel space

It has been known since the 1970s that the Torelli map $M_g \to A_g$, associating to a smooth curve its jacobian, extends to a regular map from the Deligne-Mumford compactification $\bar{M}_g$ to the 2nd Voronoi compactification $\bar{A}_g^{vor}$. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification $\bar{A}_g^{perf}$ is also regular, and moreover $\bar{A}_g^{vor}$ and $\bar{A}_g^{perf}$ share a common Zariski open neighborhood of the image of $\bar{M}_g$. We also show that the map to the Igusa monoidal transform (central cone compactification) is NOT regular for $g\ge9$; this disproves a 1973 conjecture of Namikawa.Comment: To appear in Inventiones Mathematica

Rationality of the moduli spaces of plane curves of sufficiently large degree

We prove that the moduli space of plane curves of degree d is rational for all sufficiently large d.Comment: 18 pages; 1 figure; Macaulay2 scripts used can be found at http://www.uni-math.gwdg.de/bothmer/rationality/ or at the end of the latex source fil

7-Gons and genus three hyperelliptic curves

In this paper, we will give a general but completely elementary description for hyperelliptic curves of genus three whose Jacobian varieties have endomorphisms by the real cyclotomic field ℚ(ζ7 + ζ̄7). We study the algebraic correspondences on these curves which are lifts of algebraic correspondences on a conic in P2 associated with Poncelet 7-gons. These correspondences induce endomorphisms φ on the Jacobians which satisfy φ3+φ2-2φ-1=0. Moreover, we study Humbert\u27s modular equations which characterize the curves of genus three having these real multiplications. © 2012 Springer-Verlag