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Angelo Vistoli

Publications and source records attributed to Angelo Vistoli.

At least 37 records · Page 2Linked to original sources
arXiv2016

On the dimension of the locus of determinantal hypersurfaces

The characteristic polynomial of an $r$-tuple $(A_1,..., A_r)$ of $n \times n$ matrices is the determinant $\det(x_0 I + x_1 A_1 + ... + x_r A_r)$. We show that if $r$ is at least 3 and $A = (A_1,..., A_r)$ is an $r$-tuple of matrices in general position, then up to conjugacy there are only finitely many $r$-tuples of matrices with the same characteristic polynomial as $A$. Equivalently, the locus of determinantal hypersurfaces of degree $n$ in $P^r$ is irreducible of dimension $(r-1)n^2 + 1$.

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arXiv2012

The Nori fundamental gerbe of a fibered category

We give a condition that ensures that a fibered category over a field admits a universal morphism to a profinite gerbe. This fundamental gerbe generalizes both Nori's fundamental group scheme and Deligne's relative fundamental groupoid. Using a simplified notion of essentially finite bundle, we also give a tannakian construction. As an application, we show how the fundamental gerbe enables to formulate a version of Grothendieck's section conjecture in arbitrary characteristic. We then study various natural quotients of the fundamental gerbe.

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arXiv2012

Parabolic sheaves on logarithmic schemes

We show how the natural context for the definition of parabolic sheaves on a scheme is that of logarithmic geometry. The key point is a reformulation of the concept of logarithmic structure in the language of symmetric monoidal categories, which might be of independent interest. Our main result states that parabolic sheaves can be interpreted as quasi-coherent sheaves on certain stacks of roots.

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arXiv2011

A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

We compute the essential dimension of the functors Forms_{n,d} and Hypersurf_{n, d} of equivalence classes of homogeneous polynomials in n variables and hypersurfaces in P^{n-1}, respectively, over any base field k of characteristic 0. Here two polynomials (or hypersurfaces) over K are considered equivalent if they are related by a linear change of coordinates with coefficients in K. Our proof is based on a new Genericity Theorem for algebraic stacks, which is of independent interest. As another application of the Genericity Theorem, we prove a new result on the essential dimension of the stack of (not necessarily smooth) local complete intersection curves.

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arXiv2011

Deformation theory from the point of view of fibered categories

We give an exposition of the formal aspects of deformation theory in the language of fibered categories, instead of the more traditional one of functors. The main concepts are that of tangent space to a deformation problem, obstruction theory, versal and universal formal deformations. We include proofs of two key results: a versione of Schlessinger's Theorem in this context, and the Ran--Kawamata vanishing theorem for obstructions. We accompany this with a detailed analysis of three important cases: smooth varieties, local complete intersection subschemes and coherent sheaves.

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arXiv2010

On the essential dimension of infinitesimal group schemes

We discuss essential dimension of group schemes, with particular attention to infinitesimal group schemes. We prove that the essential dimension of a group scheme of finite type over a field k is at least equal to the difference between the dimension of its Lie algebra and its dimension. Furthermore, we show that the essential dimension of a trigonalizable group scheme of length p^{n} over a field of characteristic p>0 is at most n. We give several examples.

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arXiv2010

Stacks of trigonal curves

In this paper we give a presentation of the stack of trigonal curves as a quotient stack, and we compute its Picard group.

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arXiv2010

Twisted stable maps to tame Artin stacks

This paper is a continuation of our earlier development of a theory of tame Artin stacks. Our main goal here is the construction of an appropriate analogue of Kontsevich's space of stable maps in the case where the target is a tame Artin stack. When the target is a tame Deligne--Mumford stack, the theory was developed by Abramovich and Vistoli, and found a number of applications. The theory for arbitrary tame Artin stacks developed here is very similar, but it is necessary to overcome a number of technical hurdles and to generalize a few questions of foundation.

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arXiv2009

Essential dimension of moduli of curves and other algebraic stacks

In this paper we address questions of the following type. Let k be a base field and K/k be a field extension. Given a geometric object X over a field K (e.g. a smooth curve of genus g) what is the least transcendence degree of a field of definition of $X$ over the base field k? In other words, how many independent parameters are needed to define X? To study these questions we introduce a notion of essential dimension for an algebraic stack. In particular, we give a complete answer to the question above when the geometric objects X are smooth or stable curves, and, in the appendix by N. Fakhruddin, for principally polarized abelian varieties. This paper overlaps with our earlier preprint arXiv:math/0701903 . That preprint has splintered into several parts, which have since acquired a life of their own. In particular, see "Essential dimension, spinor groups, and quadratic forms", by the same authors, and "Some consequences of the Karpenko-Merkurjev theorem", by Meyer and Reichstein (arXiv:0811.2517).

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arXiv2009

Essential dimension, spinor groups and quadratic forms

We prove that the essential dimension of the spinor group Spin_n grows exponentially with n; in particular, we give a precise formula for this essential dimension when n is not divisible by 4. We use this result to show that the number of 3-fold Pfister forms needed to represent the Witt class of a general quadratic form of rank n with trivial discriminant and Hasse-Witt invariant grows exponentially with n. This paper overlaps with our earlier preprint arXiv:math/0701903 . That preprint has splintered into several parts, which have since acquired a life of their own. In particular, see "Essential dimension of moduli of curves and other algebraic stacks", by the same authors, and "Some consequences of the Karpenko-Merkurjev theorem", by Meyer and Reichstein (arXiv:0811.2517).

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arXiv2007

Tame stacks in positive characteristic

We introduce and study a class of algebraic stacks with finite inertia in positive and mixed characteristic, which we call tame algebraic stacks. They include tame Deligne-Mumford stacks, and are arguably better behaved than general Deligne-Mumford stacks. We also give a complete characterization of finite flat linearly reductive schemes over an arbitrary base. Our main result is that tame algebraic stacks are étale locally quotient by actions of linearly reductive finite group schemes. In a subsequent paper we will show that tame algebraic stacks admit a good theory of stable maps.

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arXiv2007

Essential dimension and algebraic stacks

We define and study the essential dimension of an algebraic stack. We compute the essential dimension of the stacks Mgn and MgnBar of smooth, or stable, n-pointed curves of genus g. We also prove a general lower bound for the essential dimension of algebraic groups with a non-trivial center. Using this, we find new exponential lower bounds for the essential dimension of spin groups and new formulas for the essential dimension of some finite p-groups. Finally, we apply the lower bound for spin groups to the theory of the Witt ring of quadratic forms over a field k.

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arXiv2005

On the cohomology and the Chow ring of the classifying space of PGL_p

We investigate the integral cohomology ring and the Chow ring of the classifying space of the complex projective linear group PGL_p, when p is an odd prime. In particular, we determine its additive structure completely, and we reduce the problem of determing its multiplicative structure to a problem in invariant theory.

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arXiv2005

Birational isomorphisms between twisted group actions

Let X be an algebraic variety with a generically free action of a connected algebraic group G. Given an automorphism u of G, we will denote by X^u the same variety X with the G-action given by twisted by u. V. L. Popov asked if X and X^u are always G-equivariantly birationally isomorphic. We construct examples to show that this is not the case in general, but we prove that X and X^u are always stably birationally isomorphic, i.e., birationally isomorphic after taking products with an affine space with trivial G-action.

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arXiv2004

Higher algebraic K-theory for actions of diagonalizable groups

We study the K-theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces: our main result shows how to reconstruct the K-theory ring of such an action from the K-theory rings of the loci where the stabilizers have constant dimension. We apply this to the calculation of the equivariant K-theory of toric varieties, and give conditions under which the Merkurjev spectral sequence degenerates, so that the equivariant K-theory ring determines the ordinary K-theory ring. We also prove a very refined localization theorem for actions of this type.

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