Exceptional loci in algebraic surfaces
We study the algebraic exceptional set for surfaces (S,B) of log general type, when B has at least three irreducible components; we prove that in most cases it is finite or empty.
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arXiv subjects
Amos Turchet
Publications and source records attributed to Amos Turchet.
The algebraic Green-Griffiths-Lang conjecture for complements of very general pairs of divisors
We prove that the complement of a very general pair of hypersurfaces of total degree $2n$ in $\mathbb{P}^n$ is algebraically hyperbolic modulo a proper closed subvariety. This provides evidence towards conjectures of Lang-Vojta and Green-Griffiths, and partially extends previous work of Chen, Pacienza-Rousseau, and Chen-Riedl and the third author.
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Tropical curves of unibranch points and hypertangency
We study integral plane curves meeting at a single unibranch point and show that such curves must satisfy two equivalent conditions. A numeric condition: the local invariants of the curves at the contact point must be arithmetically related. A geometric condition: the tropical curves that we associate to the contact point must be isomorphic. Moreover, we prove closed formulas for the delta-invariant of a unibranch singularity, and for the dimension of the loci of curves with an assigned unibranch point. Our work is motivated by interest in the Lang exceptional set.
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Hypertangency of plane curves and the algebraic exceptional set
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.
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Rational distances from given rational points in the plane
In this paper we study sets of points in the plane with rational distances from r prescribed points P_1, ...,P_r. A crucial case arises for r = 3, where we provide simple necessary and sufficient conditions for the density of this set in the real topology. We show in the Main Theorem that these conditions can be checked effectively (via congruences), proving that a related class of K3 surfaces satisfies the local-global principle. In particular, these conditions are always satisfied when P_1, P_2, P_3 are rational. This result completes and goes beyond the analysis of Berry, who worked under stronger assumptions, not always fulfilled for instance in all the cases where P_1, P_2, P_3 are rational. On the other hand, for r\ge 4, we show that points with rational distances correspond to rational points in a surface of general type, hence conjecturally not Zariski dense. 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 analogues for points with distances in certain ring of integers.
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Simply connectedness and hyperbolicity
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.
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Greatest Common Divisor results on semiabelian varieties and a Conjecture of Silverman
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.
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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.
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Lang-Vojta Conjecture over function fields for surfaces dominating $\mathbb{G}_m^2$
We prove the nonsplit case of the Lang-Vojta conjecture over function fields for surfaces of log general type that are ramified covers of $\mathbb{G}_m^2$. 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 $\mathbb{P}^2$. We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.
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Divisibility of polynomials and degeneracy of integral points
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.
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Around the Chevalley-Weil Theorem
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.
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Nonspecial varieties and Generalized Lang-Vojta conjectures
We construct a family of fibered threefolds $X_m \to (S , Δ)$ 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 \emph{weakly special} but not \emph{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.
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Hyperbolicity of varieties of log general type
These notes provide an overview of various notions of hyperbolicity for varieties of log general type from the viewpoint of both arithmetic and birational geometry. The main results are based on our paper entitled "Hyperbolicity and uniformity of varieties of log general type." They are expanded notes from a minicourse the authors gave as part of the Geometry and arithmetic of orbifolds workshop at UQÁM.
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Hyperbolicity and uniformity of varieties of log general type
Projective varieties with ample cotangent bundle satisfy many notions of hyperbolicity, and one goal of this paper is to discuss generalizations to quasi-projective varieties. A major hurdle is that the naive generalization fails, i.e. the log cotangent bundle is never ample. Instead, we define a notion called almost ample which roughly asks that the log cotangent is as positive as possible. We show that all subvarieties of a quasi-projective variety with almost ample log cotangent bundle are of log general type. In addition, if one assumes globally generated then we obtain that such varieties contain finitely many integral points. In another direction, we show 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 almost ample log cotangent sheaf are uniformly bounded.
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The Erdős-Ulam problem, Lang's conjecture, and uniformity
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.
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A fibered power theorem for pairs of log general type
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.
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Fibered Threefolds and Lang-Vojta's Conjecture over Function Fields
Using the techniques introduced by Corvaja and Zannier 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.
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Invitation to Integral and Rational points on curves and surfaces
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.
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