Moduli theoretic study of Fano varieties and Enriques surfaces

作为数值平凡对合的下一种情况，我们将Entique曲面的数值反射对合分为两类。一个是E7型，我们成功地写出了它的正则椭圆原纤维的显式方程。作为与Hisanor Ohashi的共同工作，我们对Enrique曲面上具有半辛作用的所有有限群进行了分类。我们还对所有自同构群实际上是阿贝尔的Enrique曲面进行了分类。在这两种情况下，证明的关键是Enrique曲面的根系的概念。对于高亏格的K3曲面，我们利用扭转三次曲面的模空间的GPS紧化，成功地证明了亏格为16的K3曲面的模空间的单旋性。

As the next case of numerically trivial involution, we classified the numerically reflective involutions of Entiques surfaces into two types. One is of type E7, for which we succeeded to write down the explicit equations of their canonical elliptic fibrations. As a joint work with Hisanor Ohashi, we classified all finite groups which have a semi-symplectic action on an Enriques surface. We also classified all Enriques surfaces whose automorphism groups are virtually abelian. In both cases, the key of proof is the notion of root system of an Enriques surface. It was first defined by Nikulin in 80's and refined in this study of ours.As for K3 surfaces of higher genus, we succeeded in proving the unirationality of the moduli space of K3 surfaces of genus 16 using the GPS compactification of the moduli space of twisted cubics.