Embedding non-projective Mori dream space

Abstract

This paper is devoted to extend some Hu-Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, Q-factorial algebraic varieties with finitely generated class group and Cox ring, here called weak Mori dream spaces (wMDS), are considered. Conditions guaranteeing the existence of a neat embedding of a (completion of a) wMDS into a complete toric variety are studied, showing that, on the one hand, those which are complete and admitting low Picard number are always projective, hence Mori dream spaces in the sense of Hu-Keel. On the other hand, an example of a wMDS does not admitting any neat embedded sharp completion (i.e. Picard number preserving) into a complete toric variety is given, on the contrary of what Hu and Keel exhibited for a MDS.Moreover, termination of the Mori minimal model program (MMP) for every divisor and a classification of rational contractions for a complete wMDS are studied, obtaining analogous conclusions as for a MDS.Finally, we give a characterization of a wMDS arising from a small Qfactorial modification of a projective weak Q-Fano variety.

An extension of polar duality of toric varieties and its consequences in Mirror Symmetry

“…Then results by Batyrev and Cox [8] and Bunnet [15] cannot be applied to guarantee a good definition of a moduli space M Y ∨ . By the way, observe that the computation performed in Theorem 5.3 is consistent with the definition of m Y ∨ given in (46). In fact…”

“…actually Hu-Keel proved this fact when X is projective, but this hypothesis is unnecessary (see e.g. [46,Thm. 3.7] and references therein).…”

“…give rise to corresponding families of LG models {(T, f a )} and {(T a , f ∨ b )}, respectively, whose variation turns out to be "anchored at infinity" by their framing; when (suitable resolutions of) Y and Y ∨ are Calabi-Yau varieties, then these families of LG models are precisely those considered in [37], meaning that, in this case, their spaces of "anchored" versal deformations are smooth [ The action of P GL(n + 1) on P n extends naturally to an action on the parameter space P H 0 (O P n (d) . Recalling (46), define the number of complex moduli of a generic (smooth) hypersurface Y = Y d ⊂ P n , of degree d, to be the following one…”

“…In particular, every facet of ∆ −K Xa does not contain any interior point. Then, recalling (46), one find that…”