Dipartimento di Matematica e Fisica · Università degli Studi Roma 3 · L.go S. L. Murialdo 1 · 00146, ROMA

amos(dot)turchet(at)uniroma3(dot)it · Office 201 · (+39 06 5733) 8244I am a Ricercatore a tempo determinato di tipo B (tenure track) 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.

with F. Barroero and L. Capuano

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.

Arxiv -

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 -

with E. Rousseau and J. T.-Y. Wang

Math. Annalen (2023), published online - 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 -

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 -

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 -

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 -

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 -

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 -

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 -

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 -

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 -

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 -

Università di Roma Tre

- CR 510: Crittosistemi Ellittici - Moodle.
- GE 110: Geometria e Algebra Lineare 1 (esercitazioni) - Pagina Web.

- GE 220: Topologia - Moodle.
- GE 110: Geometria e Algebra Lineare 1 (esercitazioni).

- 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

- Uniform Mordell (after Dimitrov-Gao-Habegger)

Ricercatore a Tempo Determinato di tipo B (tenure track)

November 2020 - Current

Junior Visiting Position

September 2019 - October 2020

Acting Assistant Professor

September 2016 - June 2019

Postdoctoral Researcher

August 2014 - July 2016

CoAuthors

- Kenny Ascher
- Fabrizio Barroero
- Lucas Braune
- Laura Capuano
- Pietro Corvaja
- Pranabesh Das
- Kristin DeVleming
- Erwan Rousseau
- Julie Tzu-Yueh Wang
- Umberto Zannier

Useful Links

- Math ArXiv.org
- Mathscinet
- Practical suggestions for mathematical writing (Poonen)
- Practical suggestions for mathematical writing (Bell, Kadets, Srinivasan, Triantafillou, and Vogt)
- Mathematics Genealogy Project
- Lean Community
- CoCalc
- Conferences in Algebraic Geometry (by Vakil)
- Conferences in Arithmetic Geometry (by Kedlaya)
- Conferences in Number Theory
- Nordic Number Theory Network
- Winter Meeting in Algebra and Geometry
- Roman Number Theory Association
- Roma Tre Geometry Seminars