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Orsola Tommasi

Publications and source records attributed to Orsola Tommasi.

At least 19 recordsLinked to original sources
arXiv2024

Stability conditions for line bundles on nodal curves

We introduce the abstract notion of a \emph{smoothable fine compactified Jacobian} of a nodal curve, and of a family of nodal curves whose general element is smooth. Then we introduce the notion of a combinatorial stability condition for line bundles and their degenerations. We prove that smoothable fine compactified Jacobians are in bijection with these stability conditions. We then turn our attention to \emph{fine compactified universal Jacobians}, that is, fine compactified Jacobians for the moduli space $\overline{\mathcal{M}}_g$ of stable curves (without marked points). We prove that every fine compactified universal Jacobian is isomorphic to the one first constructed by Caporaso, Pandharipande and Simpson in the nineties. In particular, without marked points, there exists no fine compactified universal Jacobian unless $\gcd(d+1-g, 2g-2)=1$.

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arXiv2024

Multiplicative Chow-Künneth decomposition and homology splitting of configuration spaces

We construct a splitting of the cohomology of configuration spaces of points on a smooth proper variety with a multiplicative Chow--Künneth decomposition. Applied to hyperelliptic curves, this shows that the hyperelliptic Torelli group acts trivially on the rational cohomology of ordered configuration spaces of points. Moreover, if $H_{g,n}$ denotes the moduli space of $n$-pointed hyperelliptic curves, the Leray spectral sequence for the forgetful map $H_{g,n} \to H_g$ degenerates immediately, in sharp contrast to the forgetful map from $M_{g,n}$ to $M_g$. This allows for new detailed calculations of the cohomology of $M_{2,n}$ for $n \leq 5$, and the stable cohomology of $H_{g,n}$ for $n \leq 5$. We also give a detailed study of the cohomology of symplectic local systems on $M_2$.

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arXiv2022

Geometry of genus one fine compactified universal Jacobians

We introduce a general abstract notion of fine compactified Jacobian for nodal curves of arbitrary genus. We focus on genus 1 and prove combinatorial classification results for fine compactified Jacobians in the case of a single nodal curve and in the case of the universal family over the moduli space of stable pointed curves. We show that if the fine compactified Jacobian of a nodal curve of genus 1 can be extended to a smoothing of the curve, then it can be described as the moduli space of stable sheaves with respect to some polarisation. In the universal case we construct new examples of genus 1 fine compactified universal Jacobians. Then we give a formula for the Hodge and Betti numbers of each genus 1 fine compactified universal Jacobian and prove that their even cohomology is algebraic.

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arXiv2018

The topology of $A_g$ and its compactifications

We survey old and new results about the cohomology of the moduli space $A_g$ of principally polarized abelian varieties of genus $g$ and its compactifications. The main emphasis lies on the computation of the cohomology for small genus and on stabilization results. We review both geometric and representation theoretic approaches to the problem. The appendix provides a detailed discussion of computational methods based on trace formulae and automorphic representations, in particular Arthur's endoscopic classification of automorphic representations for symplectic groups.

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arXiv2017

Stable Betti numbers of (partial) toroidal compactifications of the moduli space of abelian varieties

We present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of ${\mathcal A}_g$, satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers in close to top degree for the perfect cone toroidal compactification. This suggests the existence of an algebra structure on the stable cohomology of the perfect cone compactification in close to top degree.

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arXiv2016

Stable cohomology of the perfect cone toroidal compactification of ${\mathcal A}_g$

We show that the cohomology of the perfect cone (also called first Voronoi) toroidal compactification of the moduli space of complex principally polarized abelian varieties stabilizes, in close to the top degree. Moreover, we show that this stable cohomology is purely algebraic, and we compute it in degree up to 13. Our explicit computations and stabilization results apply in greater generality to various toroidal compactifications and partial compactifications, and in particular we show that the cohomology of the matroidal partial compactification stabilizes (in low degree). For degree up to 8, we describe explicitly the generators of the cohomology. We also discuss various approaches to computing all of the stable cohomology in arbitrary degree.

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arXiv2014

Stable cohomology of spaces of non-singular hypersurfaces

We prove that the rational cohomology of the space of non-singular complex homogeneous polynomials of degree d in a fixed number of variables stabilizes to the cohomology of the general linear group for d sufficiently large.

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arXiv2013

The orbifold cohomology of moduli of genus 3 curves

In this work we study the additive orbifold cohomology of the moduli stack of smooth genus g curves. We show that this problem reduces to investigating the rational cohomology of moduli spaces of cyclic covers of curves where the genus of the covering curve is g. Then we work out the case of genus g=3. Furthermore, we determine the part of the orbifold cohomology of the Deligne-Mumford compactification of the moduli space of genus 3 curves that comes from the Zariski closure of the inertia stack of M_3.

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arXiv2012

The Gorenstein conjecture fails for the tautological ring of $\mathcal{\bar M}_{2,n}$

Let $N$ be the smallest integer such that there is a non-tautological cohomology class of even degree on $\mathcal{\bar M}_{2,N}$. We remark that there is such a non-tautological class on $\mathcal{\bar M}_{2,20}$, by work of Graber and Pandharipande. We show that $\mathcal{\bar M}_{2,N}$ has non-tautological cohomology only in one degree, which is not the middle degree. In particular, it follows that the tautological ring of $\mathcal{\bar M}_{2,N}$ is not Gorenstein. We present some evidence suggesting that N=20 holds.

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arXiv2012

Cohomology of the second Voronoi compactification of A_4

In this paper we compute the cohomology groups of the second Voronoi compactification of the moduli space of abelian fourfolds in all degrees with the exception of the middle degree 10. We also compute the cohomology groups of the perfect cone compactification in degree < 10. The main tool is the investigation of the strata of the compactification corresponding to semi-abelic varieties with constant torus rank.

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arXiv2009

Cohomology of the toroidal compactification of A_3

We prove that the cohomology groups with rational coefficients of the Voronoi compactification of the moduli space of abelian threefolds coincide with the Chow groups of that space, as determined by Van der Geer.

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arXiv2008

On the Debarre-de Jong and Beheshti-Starr conjectures on hypersurfaces with too many lines

We show that the Debarre-de Jong conjecture that the Fano scheme of lines on a smooth hypersurface of degree at most n in n-dimensional projective space must have its expected dimension, and the Beheshti-Starr conjecture that bounds the dimension of the Fano scheme of lines for hypersurfaces of degree at least n in n-dimensional projective space, reduce to determining if the intersection of the top Chern classes of certain vector bundles is nonzero.

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arXiv2007

The rational cohomology of \Mbar_4

We present two approaches to the study of the cohomology of moduli spaces of curves. Together, they allow us to compute the rational cohomology of the moduli space \Mbar_4 of stable complex curves of genus 4, with its Hodge structure.

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arXiv2007

Rational cohomology of M_3,2

We compute the rational cohomology of the moduli space of non-singular complex projective curves of genus 3 with two marked points.

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arXiv2003

Rational cohomology of the moduli space of genus 4 curves

We prove that the Poincare' polynomial of the moduli space of smooth genus 4 curves is 1+t^2+t^4+t^5. We show this by producing a stratification of the space, such that all strata are geometric quotients of complements of discriminants.

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arXiv2003

Some remarks on varieties with degenerate Gauss image

We consider projective varieties with degenerate Gauss image whose focal hypersurfaces are non-reduced schemes. Examples of this situation are provided by the secant varieties of Severi and Scorza varieties. The Severi varieties are moreover characterized by a uniqueness property.

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