The moduli space of Harnack curves in toric surfaces

The Moduli Space of Curves

In this section, we will give a sketch of the construction of the moduli space Mg of curves of genus g and the closely related moduli space Mg,n of n-pointed curves of genus g using two different approaches. Throughout this section we always assume that 2g − 2 + n > 0. In the first approach, we will embed curves in a projective space P using a sufficiently high power n of their dualizing sheaf....

Counting Curves on Toric Surfaces

A few years ago, Tzeng settled a remarkable conjecture of Goettsche on counting nodal curves on smooth surfaces, proving that the formulas are given by certain universal polynomials. At the same time, Ardila and Block used the tropical approach of Brugalle, Mihalkin and Fomin to count nodal curves on a certain class of (not necessarily smooth) toric surfaces, and obtained similar polynomiality ...

Log Geometry and the Moduli Space of Toric Varieties

In [2], Alexeev constructed a proper moduli space of polarized toric varieties. However, in addition to the main component containing the toric varieties, there were additional irreducible components which were tricky to eliminate in a canonical way. Later, Olsson [1] showed how adding a log structure to the toric varieties under consideration effectively restricted the problem enough to single...

On the moduli space of quintic surfaces

of the Dissertation On the moduli space of quintic surfaces

The Moduli Space of Surfaces with K