Good quotients of Mori dream spaces

Abstract

Abstract. We show that good quotients of algebraic varieties with finitely generated Cox ring have again finitely generated Cox ring.

Complex-analytic quotients of algebraic $$G$$ G -varieties

“…is finitely generated and since O X (X) * = C * , an application of [Bäk11] yields the desired result. Based on the observations made above, it is a natural question to ask whether there exists an intrinsic characterisation of those Mori dream spaces or varieties with finitely generated Cox ring that are in class Q G for some complex reductive group G. The following result yields two necessary criteria, the first one local and the second one of global nature.…”

On automorphisms of the punctual Hilbert schemes of K3 surfaces

“…(a) |Aut(S [2] )| = ∞ and |Aut(S)| < ∞. (b) The Hilbert-Chow morphism μ : S [2] → S (2) is an extremal crepant resolution such that the source S [2] is not a Mori dream space, but the target S (2) We close the introduction by recalling the following question of Alessandra Sarti.…”

“…The correspondence {P, Q} → {R, T } then defines a birational automorphism of S [2] of order 2 for each k = 1, 2,…”

“…As S k ⊂ P 3 contains no projective line, ι k is biregular, i.e., ι k ∈ Aut(S [2] ) [4,Proposition 11].…”

“…Let S be a projective K3 surface, i.e., a smooth, simply-connected projective surface S with O S (K S ) O S . Here K S is a canonical divisor of S. Then S [n] is a projective hyperkähler manifold, i.e., a smooth, simply-connected projective variety with everywhere non-degenerate global regular 2-form σ S [n] such that H 0 (S [n] , 2 S [n] ) = Cσ S [n] [5,8].…”