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At least 307 records · Page 17Linked to original sources
arXiv2006

K3-surfaces of genus 8 and varieties of sums of powers of cubic fourfolds

The main outcome of this paper is that the variety VSP(F,10) of presentations of a general cubic form F in 6 variables as a sum of 10 cubes is a smooth symplectic 4-fold obtained a deformation of the Hilbert square of a K3 surface of genus 8. After publishing it in Trans. Am. Math. Soc. 353, No.4, 1455-1468 (2001), it was noted to us by Eyal Markman that in Theorem 3.17 we conclude without proof that VSP(F,10) should be the 4-fold of lines on another cubic 4-fold. We correct this in the e-print "Addendum to K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds" (math.AG/0611533), where we establish that the general VSP(F,10) is in fact a new symplectic 4-fold different from the family of lines on a cubic 4-fold.

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arXiv1998

Complete moduli for families over semistable curves

This note is but a research announcement, summarizing and explaining results proven and detailed in forthcoming papers. When one studies families of objects over curves, and the objects are parametrized by a Deligne-Mumford stack M, then the families are equivalent to morphisms of curves into M. In order to have complete moduli for such families, one needs to compactify the stack of stable maps into M. It turns out that in the boundary, the curve must acquire extra structure and you'd better read the paper to see what that structure is. Applications to fibered surfaces (see math.AG/9804097), admissible covers, and level structures are discussed.

math.AG↗
arXiv1998

Partial resolutions of Hilbert type, Dynkin diagrams, and generalized Kummer varieties

We study the partial resolutions of singularities related to Hilbert schemes of points on an affine space. Consider a quotient of a vector space $V$ by an action of a finite group $G$ of linear transforms. Under some additional assumptions, we prove that the partial desingularization of Hilbert type is smooth only if the action of $G$ is generated by complex reflections. This is used to study the subvarieties of a Hilbert scheme of a complex torus. We show that any subvariety of a generic deformation of a Hilbert scheme of a torus is birational to a quotient of another torus by an action of a Weyl group of some semisimple Lie algebra. In Appendix, we produce counterexamples to a false theorem stated in our preprint math.AG/9801038.

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arXiv1999

Fourier transform for D-algebras

We construct a version of Fourier transform for a class of non-commutative algebras over abelian varieties which include algebras of twisted differential operators generalizing the previous construction of Laumon (alg-geom/9603004) and of the second author (alg-geom/9602023). We also construct the microlocal version of this transform and its etale localization in the framework of Kapranov's theory of NC-schemes (see math.AG/9802041).

math.AG↗
arXiv1999

Chiral de Rham complex. II

This paper is a sequel to math.AG/9803041. It consists of three parts. In the first part we give certain construction of vertex algebras which includes in particular the ones appearing in op. cit. In the second part we show how the cohomology ring $H^*(X)$ of a smooth complex variety $X$ could be restored from the correlation functions of the vertex algebra $RΓ(X;Ω^{ch}_X)$. In the third part, we prove first a useful general statement that the sheaf of loop algebras over the tangent sheaf $\Cal{T}_X$ acts naturally on $Ω^{ch}_X$ for every smooth $X$ (see §1). The Z-graded vertex algebra $H^*(X;Ω^{ch}_X)$ seems to be a quite interesting object (especially for compact $X$). In §2, we compute $H^0(CP^N;Ω^{ch}_{CP^N})$ as a module over $\hat{sl}(N+1)$.

math.AG↗
arXiv1999

Double coverings of octic arrangements with isolated singularities

In this paper we construct 206 examples of Calabi-Yau manifolds with different Euler numbers. All constructed examples are smooth models of double coverings of $P^3$ branched along an octic surface. We allow 11 types of (not necessary isolated) singularities in the branch locus. Thus we broaden the class of examples studied in math.AG/9902057. For every considered example we compute the Euler number and give a precise description of a resolution of singularities.

math.AG↗
arXiv2002

Toroidal varieties and the weak Factorization Theorem

The main goal of the present paper is two-fold. First we extend the theory of toroidal embeddings introduced by Kempf, Knudsen, Mumford and Saint-Donat to the class of toroidal varieties with stratifications (which is the main body of the paper). Second we give a proof of the following weak factorization theorem as an application and illustration of the theory: A birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blow ups and blow downs with smooth centers. Another proof of the weak factorization theorem appeared in a joint paper with Abramovich, Karu and Matsuki (math.AG/9904135) In that paper the theorem is stated and proven in general for proper algebraic and analytic spaces.

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arXiv1999

Asymptotically Locally Euclidean metrics with holonomy SU(m)

Let G be a nontrivial finite subgroup of U(m) acting freely on C^m - 0. Then C^m/G has an isolated quotient singularity at 0. Let X be a resolution of C^m/G, and g a Kahler metric on X. We say that g is Asymptotically Locally Euclidean (ALE) if it is asymptotic in a certain way to the Euclidean metric on C^m/G. In this paper we study Ricci-flat ALE Kahler metrics on X. We show that if G is a subgroup of SU(m) acting freely on C^m - 0, and X is a crepant resolution of C^m/G, then there is a unique Ricci-flat ALE Kahler metric in each Kahler class. This is proved using a version of the Calabi conjecture for ALE manifolds. We also show the metrics have holonomy SU(m). These results will be applied in the author's book ("Compact manifolds with special holonomy", to be published by OUP, 2000) to construct new examples of compact 7- and 8-manifolds with exceptional holonomy. They can also be used to describe the Calabi-Yau metrics on resolutions of a Calabi-Yau orbifold. The paper has a sequel, "Quasi-ALE metrics with holonomy SU(m) and Sp(m)", math.AG/9905043, which studies Kahler metrics on resolutions of non-isolated singularities C^m/G.

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arXiv1999

Quasi-ALE metrics with holonomy SU(m) and Sp(m)

This is the sequel to "Asymptotically Locally Euclidean metrics with holonomy SU(m)", math.AG/9905041. Let G be a subgroup of U(m), and X a resolution of C^m/G. We define a special class of Kahler metrics g on X called Quasi Asymptotically Locally Euclidean (QALE) metrics. These satisfy a complicated asymptotic condition, implying that g is asymptotic to the Euclidean metric on C^m/G away from its singular set. When C^m/G has an isolated singularity, QALE metrics are just ALE metrics. Our main interest is in Ricci-flat QALE Kahler metrics on X. We prove an existence result for Ricci-flat QALE Kahler metrics: if G is a subgroup of SU(m) and X a crepant resolution of C^m/G, then there is a unique Ricci-flat QALE Kahler metric on X in each Kahler class. This is proved using a version of the Calabi conjecture for QALE manifolds. We also determine the holonomy group of the metrics in terms of G. These results will be applied in the author's book ("Compact manifolds with special holonomy", to be published by OUP, 2000) to construct new examples of compact 7- and 8-manifolds with exceptional holonomy. They can also be used to describe the Calabi-Yau metrics on resolutions of a Calabi-Yau orbifold.

math.AG↗
arXiv1999

Fixed points of group actions and rational maps

The aim of this note is to give simple proofs of some results of Reichstein and Youssin (math.AG/9903162) about the behaviour of fixed points of finite group actions under rational maps. Our proofs work in any characteristic. We also give a short proof of the Nishimura lemma.

math.AG↗
arXiv1999

On the classification of hyperbolic root systems of the rank three. Part III

See Parts I and II in alg-geom/9711032 and alg-geom/9712033. Here we classify maximal hyperbolic root systems of the rank three having restricted arithmetic type and a generalized lattice Weyl vector $ρ$ with $ρ^2<0$ (i. e. of the hyperbolic type). We give classification of all reflective of hyperbolic type elementary hyperbolic lattices of the rank three. For elliptic (when $ρ^2>0$) and parabolic (when $ρ^2=0$) types it was done in Parts I and II. We apply the same arguments as for elliptic and parabolic types: the method of narrow parts of polyhedra in hyperbolic spaces, and class numbers of central symmetries. But we should say that for the hyperbolic type all considerations are much more complicated and required much more calculations and time. These results are important, for example, for Theory of Lorentzian Kac--Moody algebras and some aspects of Mirror Symmetry. We also apply these results to prove boundedness of families of algebraic surfaces with almost finite polyhedral Mori cone (see math.AG/9806047 about this subject).

math.AG↗
arXiv1999

Power series and p-adic algebraic closures

In an earlier preprint (math.AG/9810142) we gave an explicit description of the algebraic closure of the field of power series over a field of characteristic p, in terms of "generalized power series". In this paper, we give an analogous description of the algebraic closure of the fraction field of the Witt ring of a field of characteristic p, using an analogue of generalized power series in mixed characteristic due to Poonen.

math.AG↗
arXiv1999

Enriques Surfaces and other Non-Pfaffian Subcanonical Subschemes of Codimension 3

We give examples of subcanonical subvarieties of codimension 3 in projective n-space which are not Pfaffian, i.e. defined by the ideal sheaf of submaximal Pfaffians of an alternating map of vector bundles. This gives a negative answer to a question asked by Okonek. Walter had previously shown that a very large majority of subcanonical subschemes of codimension 3 in P^n are Pfaffian, but he left open the question whether the exceptional non-Pfaffian cases actually occur. We give non-Pfaffian examples of the principal types allowed by his theorem, including (Enriques) surfaces in P^5 in characteristic 2 and a smooth 4-fold in P^7. These examples are based on our previous work math.AG/9906170 showing that any strongly subcanonical subscheme of codimension 3 of a Noetherian scheme can be realized as a locus of degenerate intersection of a pair of Lagrangian (maximal isotropic) subbundles of a twisted orthogonal bundle.

math.AG↗
arXiv1999

The Picard Group of the Moduli of Higher Spin Curves

This article treats the Picard group of the moduli (stack) of r-spin curves and its compactification. Generalized spin curves, or r-spin curves are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because they are the subject of a remarkable conjecture of E. Witten, and because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. We generalize results of Cornalba, giving relations between many of the elements of the Picard group of the stacks. These relations are important in the proof of the genus-zero case of Witten's conjecture given in math.AG/9905034. We use these relations to show that when 2 or 3 divides r, then the Picard group of the open stack has non-zero torsion. And finally, we work out some specific examples for small values of g and r.

math.AG↗
arXiv1999

Exponential sums on A^n, III

We give two applications of our earlier work "Exponential sums on A^n, II" (math.AG/9909009). We compute the p-adic cohomology of certain exponential sums on A^n involving a polynomial whose homogeneous component of highest degree defines a projective hypersurface with at worst weighted homogeneous isolated singularities. This study was motivated by recent work of Garcia (Exponential sums and singular hypersurfaces, Manuscripta Math., v. 97 (1998), pp. 45-58). We also compute the p-adic cohomology of certain exponential sums on A^n whose degree is divisible by the characteristic.

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