Symmetries of order four on K3 surfaces

We study automorphisms of order four on K3 surfaces. The symplectic ones have been first studied by Nikulin, they are known to fix six points and their action on the K3 lattice is unique. In this paper we give a classification of the purely non-symplectic automorphisms by relating the structure of their fixed locus to their action on cohomology, in the following cases: the fixed locus contains a curve of genus g>0; the fixed locus contains at least a curve and all the curves fixed by the square of the automorphism are rational. We give partial results in the other cases. Finally, we classify non-symplectic automorphisms of order four with symplectic square.Comment: Final version, to appear in J. Math. Soc. Japa

Hypersurfaces in Mori dream spaces

Let X be a hypersurface of a Mori dream space Z. We provide necessary and sufficient conditions for the Cox ring R(X) of X to be isomorphic to R(Z)/(f), where R(Z) is the Cox ring of Z and f is a defining section for X. We apply our results to Calabi-Yau hypersurfaces of toric Fano fourfolds. Our second application is to general degree d hypersurfaces in P^n containing a linear subspace of dimension n-2.Comment: 12 page

Algebraic functions with even monodromy

Let $X$ be a compact Riemann surface of genus $g$ and $d\geq 12g+4$ be an integer. We show that $X$ admits meromorphic functions with monodromy group equal to the alternating group $A_d.