Towards birational aspects of moduli space of curves

[摘要] The moduli space of curves has proven itself a central object in algebraic geometry. The past decade has seen substantial progress in understanding its geometry. This has been spurred by a flurry of ideas from geometry (algebraic, symplectic, and differential), topology, combinatorics, and physics. One way of understanding its birational geometry is by describing its cones of ample and effective divisors and the dual notion of the Mori cone (the closed cone of curves). This thesis aims at giving a brief introduction to the moduli space of n-pointed stable curves of genus ... and some intuition into it and its structure. We do so by surveying what is currently known about the ample and the effective cones of ... , and the problem of determining the closed cone of curves ... The emphasis in this exposition lies on a partial resolution of the Fulton-Faber conjecture (the F-conjecture). Recently, some positive results were announced and the conjecture was shown to be true in a select few cases. Conjecturally, the ample cone has a very simple description as the dual cone spanned by the F-curves. Faber curves (or F-curves) are irreducible components of the locus in ... that parameterize curves with 3g - 4 + n nodes. There are only finitely many classes of F-curves. The conjecture has been verified for the moduli space of curves of small genus. The conjecture predicts that for large g, despite being of general type, ... behaves from the point of view of Mori theory just like a Fano variety. Specifically, this means that the Mori cone of curves is polyhedral, and generated by rational curves. It would be pleasantly surprising if the conjecture holds true for all cases. In the case of the effective cone of divisors the situation is more complicated. F-conjecture. A divisor on ... is ample (nef) if and only if it intersects positively (nonnegatively) all 1-dimensional strata or the F-curves . In other words, every extremal ray of the Mori cone of effective curves NE1(Mg,n) is generated by a one dimensional stratum. The main results presented here are: (i) the Mori cone ... is generated by F-curves when ...

[发布机构] Massachusetts Institute of Technology