Amos Turchet

Dipartimento di Matematica e Fisica · Università degli Studi Roma 3 · Via della Vasca Navale 84 · 00146, ROMA

amos(dot)turchet(at)uniroma3(dot)it · Office 0.12 · (+39 06 5733) 8244

I am an Associate Professor in the Dipartimento di Matematica e Fisica of Università di Roma Tre in Rome.

I am interested in and working on Diophantine, Arithmetic, Algebraic Geometry, and Number Theory.

Previously I was a Junior Visitor at Scuola Normale Superiore under the mentorship of prof. Umberto Zannier, an Acting Assistant Professor at the Department of Mathematics at University of Washington under the mentorship of prof. Bianca Viray and a Postdoctoral Researcher in the Math Department at Chalmers University of Technology under the mentorship of prof. Per Salberger.

I attained my Ph.D. in Mathematics on May 29, 2014, at the University of Udine under the supervision of prof. Pietro Corvaja with the Thesis Geometric Lang-Vojta Conjecture in the projective plane.

I organize the Geometry Seminars of the Geometry group of the Department of Mathematics and Physics at the Roma Tre University.

Publications

Hypertangency of plane curves and the algebraic exceptional set

with L. Caporaso

We investigate plane curves intersecting in at most two unibranched points to study the algebraic exceptional set appearing in standard conjectures of diophantine and hyperbolic geometry. Our first result compares the local geometry of two hypertangent curves, i.e. curves having maximal contact at one unibranched point. This is applied to fully describe the exceptional set and, more generally, the hyper-bitangency set, of a plane curve with three components.

Arxiv -

Rational distances from given rational points in the plane

with P. Corvaja and U. Zannier

In this paper we consider sets of points in the plane with rational distances from a prescribed finite set of n rational points. We show that for \(n\leq 3\), the points are dense in the real topology. On the other hand, for \(n\geq 4\), we show that they correspond to rational points in a surface of general type, hence conjecturally degenerate. However, at the present, we lack methods to prove this, given the fact that the surface is simply-connected, as we shall show. We give explicit proofs as well as describe in detail the geometry of the surfaces involved. In addition we discuss certain cases of density of points with distances in certain ring of integers.

Arxiv -

Simply connectedness and hyperbolicity

with C. Gasbarri, E. Rousseau and J. T.-Y. Wang

We generalize to arbitrary dimension our previous construction of simply connected weakly-special but not special varieties. We show that they satisfy the function field and complex analytic part of Campana's conjecture. Moreover, we give the first examples, in any dimension, of smooth simply connected nonisotrivial projective varieties of general type that satisfy the function field Lang's conjecture.

Arxiv -

Greatest Common Divisor results on semiabelian varieties and a Conjecture of Silverman

with F. Barroero and L. Capuano
Research in Number Theory Volume 10, article number 17, (2024) - doi: 10.1007/s40993-023-00494-2

A divisibility sequence is a sequence of integers {d_n} such that d_m divides d_n if m divides n. Results of Bugeaud, Corvaja, Zannier, among others, have shown that the gcd of two divisibility sequences corresponding to subgroups of the multiplicative group grows in a controlled way. Silverman conjectured that a similar behaviour should appear in many algebraic groups. We extend results by Ghioca-Hsia-Tucker and Silverman for elliptic curves and prove an analogue of Silverman's conjecture over function fields for abelian and split semiabelian varieties and some generalizations of this result. We employ tools coming from the theory of unlikely intersections as well as properties of the so-called Betti map associated to a section of an abelian scheme.

Journal -

Some examples of exceptional loci in Vojta Conjecture

In this short note we discuss the exceptional locus for the Lang-Vojta's conjecture in the case of the complement of two completely reducible hyperplane sections in a cubic surface. Using elementary methods, we show that generically the exceptional set is the union of the remaining 21 lines in the surface. We also describe examples in which the exceptional set is strictly larger.

Arxiv -

Divisibility of polynomials and degeneracy of integral points

with E. Rousseau and J. T.-Y. Wang
Math. Ann. 388 (2024), 1969-1999 - doi: 10.1007/s00208-023-02564-3

We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples of simply connected quasi-projective varieties that are pseudo-arithmetically hyperbolic. This generalizes results of Corvaja and Zannier obtained in dimension 2 to arbitrary dimension. The key input is an application of the Ru-Vojta's strategy. We also obtain the analogue results for function fields and Nevanlinna theory with the goal to apply them in a future paper in the context of Campana's conjectures.

Journal -

Around the Chevalley-Weil Theorem

with P. Corvaja and U. Zannier
Enseign. Math. 68 (2022), no. 1-2, 217–235 - doi: 10.4171/lem/1027

We present a proof of the Chevalley-Weil Theorem that is somewhat different from the proofs appearing in the literature and with somewhat weaker hypotheses, of purely topological type. We also provide a discussion of the assumptions, and an application to solutions of generalized Fermat equations, where our statement allows to simplify the original argument of Darmon and Granville.

Journal -

Nonspecial varieties and Generalized Lang-Vojta conjectures

with E. Rousseau and J. T.-Y. Wang
Forum of Mathematics, Sigma , Volume 9 , 2021 , e11 - doi: 10.1017/fms.2021.8

We construct a family of fibered threefolds X_m → (S,Δ) such that X_m has no étale cover that dominates a variety of general type but it dominates the orbifold (S,Δ) of general type. Following Campana, the threefolds X_m are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if m is big enough the threefolds X_m present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalizations of known conjectures on exceptional loci that fit into Campana’s program and prove some cases over function fields.

Journal -

Lang-Vojta Conjecture over function fields for surfaces dominating \(\mathbb{G}^2_m\)

with L. Capuano
Eur. J. Math. 8 (2022), no. 2, 573–610. - doi: 10.1007/s40879-021-00502-8

We prove the nonsplit case of the Lang-Vojta conjecture over function fields for surfaces of log general type that are ramified covers of the two dimensional torus. This extends results of Corvaja and Zannier, who proved the conjecture in the split case, and results of Corvaja and Zannier and the second author that were obtained in the case of the complement of a degree four and three component divisor in the projective plane. We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.

Journal -

The Erdös-Ulam problem, Lang's conjecture, and uniformity

with K. Ascher and L. Braune
Bull. Lond. Math. Soc. 52 (2020), issue 6, 1053-1063 - doi: 10.1112/blms.12381

A rational distance set is a subset of the plane such that the distance between any two points is a rational number. We show, assuming Lang's Conjecture, that the cardinalities of rational distance sets in general position are uniformly bounded, generalizing results of Solymosi-de Zeeuw, Makhul-Shaffaf, Shaffaf, and Tao. In the process, we give a criterion for certain varieties with non-canonical singularities to be of general type.

Journal -

Hyperbolicity and uniformity of varieties of log general type

with K. Ascher and K. DeVleming
Int. Math. Res. Not. (IMRN), Volume 2022, Issue 4, February 2022, 2532–2581 - doi: 10.1093/imrn/rnaa186

We show that all subvarieties of a quasi-projective variety with positive log cotangent bundle are of log general type. In addition, we show that smooth quasi-projective varieties with positive and globally generated log cotangent have finitely many integral points, generalizing a theorem of Moriwaki. Finally, we prove that the Lang-Vojta conjecture implies the number of stably integral points on curves of log general type, and surfaces of log general type with positive log cotangent sheaf are uniformly bounded.

Journal -

Fibered Threefolds and Lang-Vojta's Conjecture over Function Fields

Trans. Amer. Math. Soc. 369 (2017), 8537-8558. doi: 10.1090/tran/6968.

Using the techniques introduced by Corvaja and Zannier in 2008 we solve the non-split case of the geometric Lang-Vojta Conjecture for affine surfaces isomorphic to the complement of a conic and two lines in the projective plane. In this situation we deal with sections of an affine threefold fibered over a curve, whose boundary, in the natural projective completion, is a quartic bundle over the base whose fibers have three irreducible components. We prove that the image of each section has bounded degree in terms of the Euler characteristic of the base curve.

Journal -

A fibered power theorem for pairs of log general type

with K. Ascher
Algebra and Number Theory 10 (2016), no. 7, 1581–1600. doi: 10.2140/ant.2016.10.1581.

Let \(f: (X,D) \to B\) be a stably family with log canonical general fiber. We prove that, after a birational modification of the base \(\tilde{B} \to B\), there is a morphism from a high fibered power of the family to a pair of log general type. If in addition the general fiber is openly canonical, then there is a morphism from a high fibered power of the original family to a pair openly of log general type.

Journal -

Invitation to Integral and Rational points on curves and surfaces

with P. Das
Rational Points, Rational Curves, and Entire Holomorphic Curves on Projective Varieties, Contemporary Mathematics, vol. 654, Amer. Math. Soc., Providence, RI, 2015, pp. 53-73. doi: 10.1090/conm/654/13215.

This survey article is an introduction to Diophantine Geometry at a basic undergraduate level. It focuses on Diophantine Equations and the qualitative description of their solutions rather than detailed proofs.

Book -

Other

Hyperbolicity of varieties of log general type

with K. Ascher
Chapter in the book "Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces", Springer International Publishing (2020), 197-247. doi: 10.1007/978-3-030-49864-1_4.
Book -

Geometric Lang-Vojta Conjecture in \(\mathbb{P}^2\)

Ph.D. Thesis - Università degli Studi di Udine - 2014

Teaching

Università di Roma Tre
    2023 - 2024
  • CR 510: Crittosistemi Ellittici - Moodle.
  • Geometria (Ingegneria Elettronica)- Pagina Web.
    2022 - 2023
  • CR 510: Crittosistemi Ellittici - Moodle.
  • GE 110: Geometria e Algebra Lineare 1 (esercitazioni) - Pagina Web.
    2021 - 2022
  • GE 220: Topologia - Moodle.
  • GE 110: Geometria e Algebra Lineare 1 (esercitazioni).
    2020 - 2021
  • GE 450: Topologia Algebrica.
  • GE 220: Topologia (esercitazioni)
University of Washington
  • MATH 308: Matrix Algebra and Applications (multiple sections)
  • MATH 340: Abstract Linear Algebra
  • MATH 582: Diophantine Geometry of Curves (graduate course)
  • MATH 402: Introduction to Modern Algebra
Chalmers University of Technology
  • Scheme Theory (graduate course)
  • MVE 085: multivariable calculus (exercise and Matlab sessions)
  • MVE 016: calculus II (exercise and Matlab sessions)
Università degli studi di Udine
  • Matematica Discreta per Informatica (exercise sessions)
  • Matematica per Architettura (exercise sessions)
Reading Courses

Positions

Università di Roma Tre

Associate Professor
November 2023 - Current

Università di Roma Tre

Ricercatore a Tempo Determinato di tipo B (tenure track)
November 2020 - October 2023

Scuola Normale Superiore di Pisa

Junior Visiting Position
September 2019 - October 2020

University of Washington - Seattle, USA

Acting Assistant Professor
September 2016 - June 2019

Chalmers University of Technology - Göteborg, Sweden

Postdoctoral Researcher
August 2014 - July 2016

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