Moduli theoretic study of Fano varieties and Enriques surfaces

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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.