Characterization of varieties of Fano type via singularities of Cox rings

“…The statement is a direct consequence of diagram (11). For any further detail, the interested reader is referred to [1, § 3.3.2] (in particular Prop.…”

Section: Irrelevant Loci and Ideals Recall Remarks 22 (3) (4)mentioning

confidence: 99%

Embedding non-projective Mori dream space

Rossi

2019

Geom Dedicata

View full text Add to dashboard Cite

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.

show abstract

“…When

g = 1

and

n

is small, by Lemma ,

{boldM}_{X} false(r, scriptO, bolda false)

is an image of

{boldM}_{(C, q)} false(r, scriptO, a^{'} false)

for some large

boldq

. Thus, it is of Fano type by [, Corollary 1.3].

□

…”

Section: Finite Generationmentioning

confidence: 98%

“…Thus,

{boldM}_{X} false(r, scriptO, bolda false)

is obtained from

{boldM}_{X} false(r, scriptO, boldb false)

by taking several flips and blow‐downs. By [, Theorem , Corollary 1.3],

{boldM}_{X} false(r, scriptO, bolda false)

is also of Fano type. When

g = 1

and

n

is small, by Lemma ,

{boldM}_{X} false(r, scriptO, bolda false)

is an image of

{boldM}_{(C, q)} false(r, scriptO, a^{'} false)

for some large

boldq

.…”

Section: Finite Generationmentioning

confidence: 99%

Finite generation of the algebra of type A conformal blocks via birational geometry II: higher genus

Moon

Yoo

2019

Proc. London Math. Soc.

View full text Add to dashboard Cite

We prove finite generation of the algebra of type A conformal blocks over arbitrary stable curves of any genus. As an application, we construct a flat family of irreducible normal projective varieties over the moduli stack of stable pointed curves, whose fiber over a smooth curve is a moduli space of semistable parabolic bundles. This generalizes a construction of a degeneration of the moduli space of vector bundles presented in a recent work of Belkale and Gibney.

show abstract

Characterization of varieties of Fano type via singularities of Cox rings

Abstract: We show that every Mori dream space of globally Fregular type is of Fano type. As an application, we give a characterization of varieties of Fano type in terms of the singularities of their Cox rings.

Cited by 50 publications

References 30 publications

Flexible affine cones and flexible coverings

Flexible affine cones and flexible coverings

Embedding non-projective Mori dream space

Finite generation of the algebra of type A conformal blocks via birational geometry II: higher genus

Contact Info

Product

Resources

About