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Elena Martinengo

Publications and source records attributed to Elena Martinengo.

12 recordsLinked to original sources
arXiv2026

Deformations of morphisms of coherent sheaves

We generalise Hinich's Theorem of descent of Deligne groupoids to the case where the dgLas involved have no negative cohomology. We apply this result to study the infinitesimal deformations of a morphism $\alpha: {\mathcal F} \to {\mathcal G}$ of coherent sheaves, where both the sheaves $ {\mathcal F}$ and $ {\mathcal G}$ and the map $\alpha$ can be deformed, on a smooth variety over a field of characteristic zero. In particular, we provide an explicit dgLa that controls these deformations via the Deligne functor, applying the Thom-Whitney totalisation to a specific semicosimplicial dgLa, constructed from geometrical data.

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arXiv2025

Deformations of morphisms of sheaves

We analyse infinitesimal deformations of morphisms of locally free sheaves on a smooth projective variety $X$ over an algebraically closed field of characteristic zero. In particular, we describe a differential graded Lie algebra controlling the deformation problem. As an application, we study infinitesimal deformations of pairs given by a locally free sheaf and a subspace of it sections with a view towards Brill-Noether theory.

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arXiv2024

Cox rings of algebraic stacks

We give a proper definition of the multiplicative structure of the following rings: the Cox ring of invertible sheaves on a general algebraic stack; and the Cox ring of rank one reflexive sheaves on a normal and excellent algebraic stack. We show that such Cox rings always exist and establish their (non-)uniqueness in terms of an Ext-group. Moreover, we compare our definition with the classical construction of a Cox ring on a variety. Finally, we give an application to the theory of Mori dream stacks.

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arXiv2023

On the local structure of the Brill-Noether locus of locally free sheaves on a smooth variety

We study the functor $\operatorname{Def}_E^k$ of infinitesimal deformations of a locally free sheaf $E$ of $\mathcal{O}_X$-modules on a smooth variety $X$, such that at least $k$ independent sections lift to the deformed sheaf, where $h^0(E) \geq k$. We deduce some information on the $k$-th Brill-Noether locus of $E$, such as the description of the tangent cone at some singular points, of the tangent space at some smooth ones and some links between the smoothness of the functor $\operatorname{Def}_E^k$ and the smoothness of some well know deformations functors and their associated moduli spaces. As a tool for the investigation of $\operatorname{Def}_E^k$, we study infinitesimal deformations of the pairs $(E,U)$, where $U$ is a linear subspace of sections of $E$. We generalise to the case where $E$ has any rank and $X$ any dimension many classical results concerning the moduli space of coherent systems, like the description of its tangent space and the link between its smoothness and the injectivity of the Petri map.

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arXiv2019

On degeneracy loci of equivariant bi-vector fields on a smooth toric variety

We study equivariant bi-vector fields on a toric variety. We prove that, on a smooth toric variety of dimension $n$, the locus where the rank of an equivariant bi-vector field is $\leq 2k$ is not empty and has at least a component of dimension $\geq 2k+1$, for all integers $k> 0$ such that $2k < n$. The same is true also for $k=0$, if the toric variety is smooth and compact. While for the non compact case, the locus in question has to be assumed to be non empty.

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arXiv2018

Maps of Mori dream spaces

Given a map $ϕ: X \to Y$ of $\mathbb Q$-factorial Mori dream spaces, one can ask whether this map is induced by a homogeneous homomorphism $R(Y) \to R(X)$ of Cox rings. As soon as $Y$ is singular, such a homomorphism needs not to exist, as pulling back Weil divisors is not well-defined. In this article, we prove that there is a unique Cox lift $Φ: \mathcal X \to \mathcal Y$ of Mori dream stacks coming from a homogeneous homomorphism $R(Y) = R(\mathcal Y) \to R(\mathcal X)$, where $\mathcal Y$ is a canonical stack to $Y$ and $\mathcal X$ is obtained from $X$ by root constructions. Moreover, $ϕ$ is induced from $Φ$ by passing to coarse moduli spaces.

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arXiv2015

Mori Dream Stacks

We propose a generalisation of Mori dream spaces to stacks. We show that this notion is preserved under root constructions and taking abelian gerbes. Unlike the case of Mori dream spaces, such a stack is not always given as a quotient of the spectrum of its Cox ring by the Picard group. We give a criterion when this is true in terms of Mori dream spaces and root constructions. Finally, we compare this notion with the one of smooth toric Deligne-Mumford stacks.

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arXiv2012

A short note on infinity-groupoids and the period map for projective manifolds

A common criticism of infinity-categories in algebraic geometry is that they are an extremely technical subject, so abstract to be useless in everyday mathematics. The aim of this note is to show in a classical example that quite the converse is true: even a naive intuition of what an infinity-groupoid should be clarifies several aspects of the infinitesimal behaviour of the periods map of a projective manifold. In particular, the notion of Cartan homotopy turns out to be completely natural from this perspective, and so classical results such as Griffiths' expression for the differential of the periods map, the Kodaira principle on obstructions to deformations of projective manifolds, the Bogomolov-Tian-Todorov theorem, and Goldman-Millson quasi-abelianity theorem are easily recovered.

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arXiv2010

Semicosimplicial DGLAs in deformation theory

We identify Cech cocycles in nonabelian (formal) group cohomology with Maurer-Cartan elements in a suitable L-infinity algebra. Applications to deformation theory are described.

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arXiv2010

Infinitesimal deformations of Hitchin pairs and Hitchin map

We identify dglas that control infinitesimal deformations of the pairs (manifold, Higgs bundle) and of Hitchin pairs. As a consequence, we recover known descriptions of first order deformations and we refine known results on obstructions. Secondly we prove that the Hitchin map is induced by a natural L-infinity morphism and, by standard facts about L-infinity algebras, we obtain new conditions on obstructions to deform Hitchin pairs.

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arXiv2009

Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves

We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf F are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf End(E), where E is any locally free resolution of F. In particular, one recovers the well known fact that the tangent space to deformations of F is Ext^1(F,F), and obstructions are contained in Ext^2(F,F). The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA whose cohomology is concentrated in nonnegative degrees with a noncommutative Cech cohomology-type functor.

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