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Tommaso de Fernex

Publications and source records attributed to Tommaso de Fernex.

At least 19 recordsLinked to original sources
arXiv2026

Relative Mather discrepancy on arc spaces

Given any generically étale morphism of varieties $f \colon X \to Y$, we define the relative Mather discrepancy function on the arc space $X_\infty$ of the domain and show that this function computes the dimension of the kernel of the differential map of the induced morphism on arc spaces $f_\infty \colon X_\infty \to Y_\infty$. We relate this result to the change-of-variable formula in motivic integration. We introduce the notion of $\widehat K$-equivalence, which agrees with $K$-equivalence for smooth varieties, and prove that $\widehat K$-equivalent varieties of arbitrary characteristic define the same class in the motivic ring.

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arXiv2024

Families of jets of arc type and higher (co)dimensional Du Val singularities

Families of jets through singularities of algebraic varieties are here studied in relation to the families of arcs originally studied by Nash. After proving a general result relating them, we look at normal locally complete intersection varieties with rational singularities and focus on a class of singularities we call "higher Du Val singularities", a higher dimensional (and codimensional) version of Du Val singularities that is closely related to Arnold singularities. More generally, we introduce the notion of "higher compound Du Val singularities", whose definition parallels that of compound Du Val singularities. For such singularities, we prove that there exists a one-to-one correspondence between families of arcs and families of jets of sufficiently high order through the singularities. In dimension two, the result partially recovers a theorem of Mourtada on the jet schemes of Du Val singularities. As an application, we give a solution of the Nash problem for higher Du Val singularities.

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arXiv2023

Motivic integration on Berkovich spaces

We define a motivic measure on the Berkovich analytification of an algebraic variety defined over a trivially valued field, and introduce motivic integration in this setting. The construction is geometric with a similar spirit as Kontsevich's original definition, and leads to the formulation of a functorial theory which mirrors, in this aspect, the approach of Cluckers and Loeser via constructibe motivic functions. A version of the integral over nontrivially valued fields and its relation to Hrushovski and Kazhdan's integration are also discussed.

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arXiv2023

Grothendieck--Lefschetz for ample subvarieties

We establish a Grothendieck--Lefschetz theorem for smooth ample subvarieties of smooth projective varieties over an algebraically closed field of characteristic zero and, more generally, for smooth subvarieties whose complement has small cohomological dimension. A weaker statement is also proved in a more general context and in all characteristics. Several applications are included.

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arXiv2023

On arc fibers of morphisms of schemes

Given a morphism $f \colon X \to Y$ of schemes over a field, we prove several finiteness results about the fibers of the induced map on arc spaces $f_\infty \colon X_\infty \to Y_\infty$. Assuming that $f$ is quasi-finite and $X$ is separated and quasi-compact, our theorem states that $f_\infty$ has topologically finite fibers of bounded cardinality and its restriction to $X_\infty \setminus R_\infty$, where $R$ is the ramification locus of $f$, has scheme-theoretically finite reduced fibers. We also provide an effective bound on the cardinality of the fibers of $f_\infty$ when $f$ is a finite morphism of varieties over an algebraically closed field, describe the ramification locus of $f_\infty$, and prove a general criterion for $f_\infty$ to be a morphism of finite type. We apply these results to further explore the local structure of arc spaces. One application is that the local ring at a stable point of the arc space of a variety has finitely generated maximal ideal and topologically Noetherian spectrum, something that should be contrasted with the fact that these rings are not Noetherian in general; a lower-bound to the dimension of these rings is also obtained. Another application gives a semicontinuity property for the embedding dimension and embedding codimension of arc spaces which extends to this setting a theorem of Lech on Noetherian local rings and translates into a semicontinuity property for Mather log discrepancies. Other applications are discussed in the paper.

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arXiv2021

Embedding codimension of the space of arcs

We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties, and study in detail the case of arc spaces of schemes of finite type over a field. Viewing the embedding codimension as a measure of singularities, our main result can be interpreted as saying that the singularities of the arc space are maximal at the arcs that are fully embedded in the singular locus of the underlying scheme, and progressively improve as we move away from said locus. As an application, we complement a theorem of Drinfeld, Grinberg, and Kazhdan on formal neighborhoods in arc spaces by providing a converse to their theorem, an optimal bound for the embedding codimension of the formal model appearing in the statement, a precise formula for the embedding dimension of the model constructed in Drinfeld's proof, and a geometric meaningful way of realizing the decomposition stated in the theorem.

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arXiv2020

Extending rationally connected fibrations from ample subvarieties

Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply this result in the context of Fano fibrations and prove a classification theorem for projective bundle and quadric fibration structures on ample subvarieties.

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arXiv2019

Differentials on the arc space

The paper provides a description of the sheaves of Kähler differentials of the arc space and jet schemes of an arbitrary scheme where these sheaves are computed directly from the sheaf of differentials of the given scheme. Several applications on the structure of arc spaces are presented.

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arXiv2019

Smooth solutions to the complex Plateau problem

Building on work of Du, Gao, and Yau, we give a characterization of smooth solutions, up to normalization, of the complex Plateau problem for strongly pseudoconvex Calabi--Yau CR manifolds of dimension $2n-1 \ge 5$ and in the hypersurface case when $n=2$. The latter case was completely solved by Yau for $n \ge 3$ but only partially solved by Du and Yau for $n=2$. As an application, we determine the existence of a link-theoretic invariant of normal isolated singularities that distinguishes smooth points from singular ones.

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arXiv2019

Nash blow-ups of jet schemes

Given an arbitrary projective birational morphism of varieties, we provide a natural and explicit way of constructing relative compactifications of the maps induced on the main components of the jet schemes. In the case the morphism is the Nash blow-up of a variety, such relative compactifications are shown to be given by the Nash blow-ups of the main components of the jet schemes.

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arXiv2018

Jet closures and the local isomorphism problem

If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow that the morphism is an isomorphism? This problem is called the local isomorphism problem. In this paper, we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem. This approach leads to a partial solution of the local isomorphism problem, which is shown to have a negative answer in general and a positive one in several situations of geometric interest.

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arXiv2017

Towards a link theoretic characterization of smoothness

A theorem of Mumford states that, on complex surfaces, any normal isolated singularity whose link is diffeomorphic to a sphere is actually a smooth point. While this property fails in higher dimensions, McLean asks whether the contact structure that the link inherits from its embedding in the variety may suffice to characterize smooth points among normal isolated singularities. He proves that this is the case in dimension 3. In this paper, we use techniques from birational geometry to extend McLean's result to a large class of higher dimensional singularities. We also introduce a more refined invariant of the link using CR geometry, and conjecture that this invariant is strong enough to characterize smoothness in full generality.

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arXiv2016

Birational rigidity of singular Fano hypersurfaces

We establish birational superrigidity for a large class of singular projective Fano hypersurfaces of index one. In the special case of isolated singularities, our result applies for instance to: (1) hypersurfaces with semi-homogeneous singularities of multiplicity asymptotically bounded by twice the square root of the dimension of the hypersurface, (2) hypersurfaces with isolated singularities whose Tyurina numbers satisfy a similar bound, and (3) hypersurfaces with isolated singularities whose dual variety is a hypersurface of degree sufficiently close to the expected degree.

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arXiv2015

Birationally rigid hypersurfaces

We prove that for N greater than or equal to 4, all smooth hypersurfaces of degree N in P^N are birationally superrigid. First discovered in the case N = 4 by Iskovskikh and Manin in a work that started this whole direction of research, this property was later conjectured to hold in general by Pukhlikov. The proof relies on the method of maximal singularities in combination with a delicate formula on restrictions of multiplier ideals.

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arXiv2015

Erratum to: Birationally rigid hypersurfaces

This note points out a gap in the proof of the main theorem of the article "Birationally rigid hypersurfaces" published in Invent. Math. 192 (2013), 533-566, and provides a new proof of the theorem.

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