Damián Gvirtz
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View article: Surfaces defined by pairs of polynomials
Surfaces defined by pairs of polynomials Open
We compute the Brauer group of surfaces defined by equating two bilinear forms of the same degree, assuming these forms are, in an explicit sense, sufficiently general. Our method uses a topological deformation argument and does not requir…
View article: Non-thin rational points for elliptic K3 surfaces
Non-thin rational points for elliptic K3 surfaces Open
We prove that elliptic K3 surfaces over a number field which admit a second elliptic fibration satisfy the potential Hilbert property. Equivalently, the set of their rational points is not thin after a finite extension of the base field. F…
View article: Surfaces defined by pairs of polynomials
Surfaces defined by pairs of polynomials Open
We prove that the Brauer group of the generic diagonal surface of arbitrary degree is trivial. The same method is applied to surfaces whose equation can be written as the sum of two bilinear forms. This uses a general criterion for the tri…
View article: Rational curves and the Hilbert Property on Jacobian Kummer varieties
Rational curves and the Hilbert Property on Jacobian Kummer varieties Open
A conjecture by Corvaja and Zannier predicts that smooth, projective, simply connected varieties over a number field with Zariski dense set of rational points have the Hilbert Property; this was proved by Demeio for Kummer surfaces which a…
View article: Perfectoid covers of abelian varieties
Perfectoid covers of abelian varieties Open
For an abelian variety A over an algebraically closed non-archimedean field of residue characteristic p, we show that there exists a perfectoid space which is the tilde-limit of lim←−−[p]A. Our proof also works for the larger class of abel…
View article: A Hilbert Irreducibility Theorem for Enriques surfaces
A Hilbert Irreducibility Theorem for Enriques surfaces Open
We define the over-exceptional lattice of a minimal algebraic surface of Kodaira dimension 0. Bounding the rank of this object, we prove that a conjecture by Campana and Corvaja--Zannier holds for Enriques surfaces, as well as K3 surfaces …
View article: Arithmetic surjectivity for zero-cycles
Arithmetic surjectivity for zero-cycles Open
Let f : X → Y be a proper, dominant morphism of smooth varieties over a number field k. When is it true that for almost all places v of k, the fibre XP over any point P ∈ Y (kv) contains a zero-cycle of degree 1? We develop a necessary and…
View article: Quantitative arithmetic of diagonal degree $2$ K3 surfaces
Quantitative arithmetic of diagonal degree $2$ K3 surfaces Open
In this paper we study the existence of rational points for the family of K3 surfaces over $\mathbb{Q}$ given by $$w^2 = A_1x_1^6 + A_2x_2^6 + A_3x_3^6.$$ When the coefficients are ordered by height, we show that the Brauer group is almost…
View article: Topics in the arithmetic of hypersurfaces and K3 surfaces
Topics in the arithmetic of hypersurfaces and K3 surfaces Open
This thesis is a collection of various results related to the arithmetic of K3 surfaces and hypersurfaces which were obtained by the author during the course of his PhD studies. The first part is related to Artin's conjecture on hypersurfa…
View article: Cohomology and the Brauer groups of diagonal surfaces
Cohomology and the Brauer groups of diagonal surfaces Open
We present a method for calculating the Brauer group of a surface given by a diagonal equation in the projective space. For diagonal quartic surfaces with coefficients in Q we determine the Brauer groups over Q and Q(i).
View article: Mazur's Conjecture and An Unexpected Rational Curve on Kummer Surfaces and their Superelliptic Generalisations
Mazur's Conjecture and An Unexpected Rational Curve on Kummer Surfaces and their Superelliptic Generalisations Open
We prove the following special case of Mazur's conjecture on the topology of rational points. Let $E$ be an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$. For a class of elliptic pencils which are quadratic twists of $E$ by qu…
View article: Arithmetic Surjectivity for Zero-Cycles
Arithmetic Surjectivity for Zero-Cycles Open
Let $f:X\to Y$ be a proper, dominant morphism of smooth varieties over a number field $k$. When is it true that for almost all places $v$ of $k$, the fibre $X_P$ over any point $P\in Y(k_v)$ contains a zero-cycle of degree $1$? We develop …
View article: Mazur's conjecture and an unexpected rational curve on Kummer surfaces and their superelliptic generalisations
Mazur's conjecture and an unexpected rational curve on Kummer surfaces and their superelliptic generalisations Open
We prove the following special case of Mazur’s conjecture on the topology of rational points. Let E be an elliptic curve over Q with j-invariant 1728. For a class of elliptic pencils which are quadratic twists of E by quartic polynomials, …
View article: Perfectoid covers of abelian varieties
Perfectoid covers of abelian varieties Open
For an abelian variety $A$ over an algebraically closed non-archimedean field of residue characteristic $p$, we show that there exists a perfectoid space which is the tilde-limit of $\varprojlim_{[p]}A$. Our proof also works for the larger…
View article: Division algebras and maximal orders for given invariants
Division algebras and maximal orders for given invariants Open
Brauer classes of a global field can be represented by cyclic algebras. Effective constructions of such algebras and a maximal order therein are given for $\mathbb{F}_{q}(t)$ , excluding cases of wild ramification. As part of the construct…