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We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison–Kawamata cone conjecture holds for these nef cones.
We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show the existence of purely inseparable sections with explicit bounded degree. To prove these results, we study log del Pezzo surfaces defined over imperfect fields.
In this paper, we establish a vanishing theorem of Nadel type for the Witt multiplier ideals on threefolds over perfect fields of characteristic larger than five. As an application, if a projective normal threefold over
is not klt and its canonical divisor is anti-ample, then the number of the rational points on the klt-locus is divisible by
We develop some foundational results in a higher-dimensional foliated Mori theory, and show how these results can be used to prove a structure theorem for the Kleiman–Mori cone of curves in terms of the numerical properties of
for rank 2 foliations on threefolds. We also make progress toward realizing a minimal model program (MMP) for rank 2 foliations on threefolds.
We construct non-archimedean SYZ (Strominger–Yau–Zaslow) fibrations for maximally degenerate Calabi–Yau varieties, and we show that they are affinoid torus fibrations away from a codimension-two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt (divisorially log terminal) models along one-dimensional strata.
We generalise Simpson’s nonabelian Hodge correspondence to the context of projective varieties with Kawamata log terminal (klt) singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety
and any resolution of singularities, any vector bundle on the resolution that appears to come from
numerically, does indeed come from
. Furthermore, and of independent interest, a new restriction theorem for semistable Higgs sheaves defined on the smooth locus of a normal, projective variety is established.
We bring examples of toric varieties blown up at a point in the torus that do not have finitely generated Cox rings. These examples are generalizations of our earlier work, where toric surfaces of Picard number 1 were studied. In this article we consider toric varieties of higher Picard number and higher dimension. In particular, we bring examples of weighted projective 3-spaces blown up at a point that do not have finitely generated Cox rings.
We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa’s conjecture on exponential sums, with the log canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for nonrational singularities by comparing the log canonical threshold with a local notion of the motivic oscillation index.
In this article, we prove a finiteness result on the number of log minimal models for 3-folds in
. We then use this result to prove a version of Batyrev’s conjecture on the structure of nef cone of curves on 3-folds in characteristic
. We also give a proof of the same conjecture in full generality in characteristic 0. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor
of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on
We give new estimates of lengths of extremal rays of birational type for toric varieties. We can see that our new estimates are the best by constructing some examples explicitly. As applications, we discuss the nefness and pseudo-effectivity of adjoint bundles of projective toric varieties. We also treat some generalizations of Fujita’s freeness and very ampleness for toric varieties.
We will develop a theory of multi-pointed non-commutative deformations of a simple collection in an abelian category, and construct relative exceptional objects and relative spherical objects in some cases. This is inspired by a work by Donovan and Wemyss.
be the field of fractions of a local Henselian discrete valuation ring
of characteristic zero with perfect residue field
. Assuming potential semi-stable reduction, we show that an unramified Galois action on the second
-adic cohomology group of a K3 surface over
implies that the surface has good reduction after a finite and unramified extension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga–Satake Abelian varieties. On our way, we settle existence and termination of certain flops in mixed characteristic, and study group actions and their quotients on models of varieties.
be a stable Chern character on
, and let
be the moduli space of Gieseker semistable sheaves on
with Chern character
. In this paper, we provide an approach to computing the effective cone of
. We find Brill–Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of
which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on
Using the geometry of an almost del Pezzo threefold, we show that the moduli space
one-pointed ineffective spin hyperelliptic curves is rational for every
We show that any Kawamata log terminal del Pezzo surface over an algebraically closed field of large characteristic is globally
-regular or it admits a log resolution which lifts to characteristic zero. As a consequence, we prove the Kawamata–Viehweg vanishing theorem for klt del Pezzo surfaces of large characteristic.
We prove that one can run the log minimal model program for log canonical 3-fold pairs in characteristic
. In particular, we prove the cone theorem, contraction theorem, the existence of flips and the existence of log minimal models for pairs with log divisor numerically equivalent to an effective divisor. These follow from our main results, which are that certain log minimal models are good.
-Fano pair. We give an upper bound for the volume
. This bound is essentially sharp for
. The main idea is to analyze the covering families of tigers constructed in J. McKernan (Boundedness of log terminal fano pairs of bounded index, preprint, 2002, arXiv:0205214). Existence of an upper bound for volumes is related to the Borisov–Alexeev–Borisov Conjecture, which asserts boundedness of the set of
-Fano varieties of a given dimension