Higher Dimensional Varieties Research Articles

Sharp bounds for the number of rational points on algebraic curves and dimension growth, over all global fields

AbstractLet be an algebraic curve over a number field , and denote by the degree of over . We prove that the number of ‐rational points of height at most in is bounded by where are absolute constants. We also prove analogous results for global fields in positive characteristic, and, for higher dimensional varieties. The quadratic dependence on in the bound as well as the exponent of are optimal; the novel aspect is the quadratic dependence on which answers a question raised by Salberger. We derive new results on Heath‐Brown and Serre's dimension growth conjecture for global fields, which generalize in particular the results by the first two authors and Novikov from the case . The proofs however are of a completely different nature, replacing the real analytic approach previously used by the ‐adic determinant method. The optimal dependence on is achieved by a key improvement in the treatment of high multiplicity points on mod reductions of algebraic curves.

On pseudo-effective cones of projective bundles and volume function

In this paper, we compute the pseudo-effective cones of various projective bundles [Formula: see text] over higher dimensional varieties [Formula: see text] under some assumptions on [Formula: see text] as well as on the vector bundle [Formula: see text]. We also compute the volume function on fiber product [Formula: see text] of two projective bundles over a smooth irreducible complex projective curve [Formula: see text]. In particular, we show that the volume function on the fiber product of two ruled surfaces is of polynomial type.

Sectorial Equidistribution of the Roots of x2 + 1 Modulo Primes

Abstract The equation $x^2 + 1 = 0\mod p$ has solutions whenever p = 2 or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. The roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec. The angles associated to the representation of such prime as a sum of squares are equidistributed is a famous theorem of Hecke. We give a natural way to associate between roots and angles and prove that the joint equidistribution of the sequence of pairs of roots and angles is equidistributed as well. Our approach involves an automorphic interpretation, which reduces the problem to the study of certain Poincare series on an arithmetic quotient of $SL_2(\mathbb{R})$. Since our Poincare series have a nontrivial dependence on their Iwasawa θ-coordinate, they do not factor into functions on the upper half plane, as in the case studied by Duke et al. Spectral analysis on these higher dimensional varieties involves the nonspherical spectrum, making this paper the first complete study of a nonspherical equidistribution problem, with an arithmetic application. A couple of notable challenges we had to overcome were that of obtaining pointwise bounds for nonspherical Eisenstein series and utilizing a non-spherical analogue of the Selberg inversion formula, which we believe may have further implications beyond this work.

Fields of moduli and the arithmetic of tame quotient singularities

Given a perfect field $k$ with algebraic closure $\overline {k}$ and a variety $X$ over $\overline {k}$, the field of moduli of $X$ is the subfield of $\overline {k}$ of elements fixed by field automorphisms $\gamma \in \operatorname {Gal}(\overline {k}/k)$ such that the Galois conjugate $X_{\gamma }$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset \overline {k}$ such that $X$ descends to $k'$. In this paper, we extend the formalism and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher-dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname {Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli.

On the $\lambda$-invariant of Selmer Groups Arising from Certain Quadratic Twists of Gross Curves

Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, and we write $\mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_\infty$ be the unique $\mathbb{Z}_2$-extension of $K$ unramified outside $\mathfrak{p}$. For certain quadratic and biquadratic extensions $\mathfrak{F}/K$, we prove a simple exact formula for the $\lambda$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $\mathfrak{p}$ of the field $\mathfrak{F}_\infty = \mathfrak{F} K_\infty$. Equivalently, our result determines the exact $\mathbb{Z}_2$-corank of certain Selmer groups over $\mathfrak{F}_\infty$ of a large family of quadratic twists of the higher dimensional abelian variety with complex multiplication, which is the restriction of scalars to $K$ of the Gross curve with complex multiplication defined over the Hilbert class field of $K$. We also exhibit computations of the associated Selmer groups over $K_n$ in the case when the $\lambda$-invariant is equal to $1$; here $K_n$ denotes the $n$-th layer of $K_\infty/K$.

Defects of codes from higher dimensional algebraic varieties

Defects of codes from higher dimensional algebraic varieties

Differential codes on higher dimensional varieties via Grothendieck's residue symbol

We give a new construction of linear codes over finite fields on higher dimensional varieties using Grothendieck's theory of residues. This generalizes the construction of differential codes over curves to varieties of higher dimensions.

On higher dimensional extremal varieties of general type

Relations among fundamental invariants play an important role in algebraic geometry. It is known that an $n$-dimensional variety of general type whose image of its canonical map is of maximal dimension, satisfies $\textrm{Vol} \geq 2 (p_g - n)$. In this article, we investigate the very interesting extremal situation of varieties with $\textrm{Vol} = 2(p_{g} - n)$, which we call Horikawa varieties for they are natural higher dimensional analogues of Horikawa surfaces. We obtain a structure theorem for Horikawa varieties and explore their pluriregularity. We use this to prove optimal results on projective normality of pluricanonical linear systems. We study the fundamental groups of Horikawa varieties, showing that they are simply connected. We prove results on deformations of Horikawa varieties, whose implications on the moduli space make them the higher dimensional analogue of curves of genus 2. Even though there are infinitely many families of Horikawa varieties in any given dimension $n$, we show that when the image of the canonical map is singular, the geometric genus of the Horikawa varieties is bounded by $n + 4$.

Abelian varieties with prescribed embedding and full embedding degrees

We study the problem of the embedding degree of an abelian variety over a finite field and extend some of the results of [1]. In particular, we show that for a prescribed CM field L of degree ≥4, prescribed integers m, n and a prescribed prime ℓ≡1(modm⋅n) that splits completely in L, there exists an ordinary abelian variety over a prime finite field with endomorphism algebra L, embedding degree n with respect to ℓ and full embedding degree m⋅n with respect to ℓ. We also study a class of absolutely simple higher dimensional abelian varieties whose endomorphism algebras are central over imaginary quadratic fields.

Error-Correcting Codes on Projective Bundles over Deligne–Lusztig Varieties

The aim of this article is to give lower bounds on the parameters of algebraic geometric error-correcting codes constructed from projective bundles over Deligne–Lusztig surfaces. The methods based on an intensive use of the intersection theory allow us to extend the codes previously constructed from higher-dimensional varieties, as well as those coming from curves. General bounds are obtained for the case of projective bundles of rank 2 over standard Deligne–Lusztig surfaces, and some explicit examples coming from surfaces of type A2 and 2A4 are given.

On the triviality of a family of linear hyperplanes

Let k be a field, m a positive integer, V an affine subvariety of Am+3 defined by a linear relation of the form x1r1⋯xmrmy=F(x1,…,xm,z,t), A the coordinate ring of V and G=X1r1⋯XmrmY−F(X1,…,Xm,Z,T). In [13], the second author had studied the case m=1 and had obtained several necessary and sufficient conditions for V to be isomorphic to the affine 3-space and G to be a coordinate in k[X1,Y,Z,T].In this paper, we study the general higher-dimensional variety V for each m⩾1 and obtain analogous conditions for V to be isomorphic to Am+2 and G to be a coordinate in k[X1,…,Xm,Y,Z,T], under a certain hypothesis on F. Our main theorem immediately yields a family of higher-dimensional linear hyperplanes for which the Abhyankar-Sathaye Conjecture holds.We also describe the isomorphism classes and automorphisms of integral domains of the type A under certain conditions. These results show that for each d⩾3, there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension d in positive characteristic.

A Riemann-Hurwitz-Plücker formula

We prove a simultaneous generalization of the classical Riemann-Hurwitz and Plücker formulas, addressing the total number of inflection points of a morphism from a (smooth, projective) curve to an arbitrary (smooth, projective) higher-dimensional variety. Our definition of inflection point is relative to an algebraic family of divisors on the target variety, and our formula is obtained using the theory of localized top Chern classes. In assigning multiplicities to inflection points, we frequently have to consider excess degeneracy loci, but we are able to show nonetheless that the multiplicities are always nonnegative, and are positive under very mild hypotheses.

Explicit Methods in Number Theory

The series of Oberwolfach meetings on ‘Explicit methods in number theory’ brings together people attacking key problems in number theory via techniques involving concrete or computable descriptions. Here, number theory is interpreted broadly, including algebraic and analytic number theory, Galois theory and inverse Galois problems, arithmetic of curves and higherdimensional varieties, zeta and L -functions and their special values, modular forms and functions. The 2021 meeting featured a seven-lecture minicourse on the distribution of class groups and Selmer groups. The other talks covered a broad range of topics in number theory ranging, for instance, from deterministic integer factorisation to the inverse Galois problem, rational points, and integrality of instanton numbers.

On the Bogomolov–Gieseker Inequality in Positive Characteristic

Abstract We prove a version of the Bogomolov–Gieseker (BG) inequality on smooth projective surfaces of general type in positive characteristic. Our result is stronger than a similar theorem by Langer for vector bundles of sufficiently large rank. Our inequality enables us to construct Bridgeland stability conditions with the full support property on all smooth projective surfaces in positive characteristic. We also prove the BG-type inequality for higher dimensional varieties.

Quantization on algebraic curves with Frobenius-projective structure

In the present paper, we study the relationship between deformation quantizations and Frobenius-projective structures defined on an algebraic curve in positive characteristic. A Frobenius-projective structure is an analogue of a complex projective structure on a Riemann surface, which was introduced by Y. Hoshi. Such an additional structure has some equivalent objects, e.g., a dormant \(\mathrm {PGL}_2\)-oper and a projective connection having a full set of solutions. The main result of the present paper provides a canonical construction of a Frobenius-constant quantization on the cotangent space minus the zero section on an algebraic curve by means of a Frobenius-projective structure. It may be thought of as a positive characteristic analogue of a result by D. Ben-Zvi and I. Biswas. Finally, this result generalizes to higher-dimensional varieties, as proved by I. Biswas in the complex case.

Tamagawa numbers of elliptic curves with torsion points

Let K be a global field and let E/K be an elliptic curve with a K-rational point of prime order p. In this paper, we are interested in how often the (global) Tamagawa number c(E/K) of E/K is divisible by p. This is a natural question to consider in view of the fact that the fraction $$c(E/K)/ |E(K)_{\text {tors}}|$$ appears in the second part of the Birch and Swinnerton-Dyer conjecture. We focus on elliptic curves defined over global fields, but we also prove a result for higher dimensional abelian varieties defined over $${\mathbb {Q}}$$ .

Geometry of certain Brill–Noether locus on a very general sextic surface and Ulrich bundles

Let \(X \subset {\mathbb {P}}^3\) be a very general sextic surface over complex numbers. In this paper, we study certain Brill–Noether problems for moduli of rank 2 stable bundles on X and its relation with rank 2 weakly Ulrich and Ulrich bundles. In particular, we show the non-emptiness of certain Brill–Noether loci and using the geometry of the moduli and the notion of the Petri map on higher dimensional varieties, we prove the existence of components of expected dimension. We also give sufficient conditions for the existence of rank 2 weakly Ulrich bundles \({\mathcal {E}}\) on X with \(c_1({\mathcal {E}}) =5H\) and \(c_2 \ge 91\) and partially address the question of whether these conditions really hold. We then study the possible implication of the existence of an weakly Ulrich bundle in terms of non-emptiness of Brill–Noether loci. Finally, using the existence of rank 2 Ulrich bundles on X we obtain some more non-empty Brill–Noether loci and investigate the possibility of existence of higher rank simple Ulrich bundles on X.

Functional transcendence for the unipotent Albanese map

We prove a certain transcendence property of the unipotent Albanese map of a smooth variety, conditional on the Ax-Schanuel conjecture for variations of mixed Hodge structure. We show that this property allows the Chabauty-Kim method to be generalized to higher-dimensional varieties. In particular, we conditionally generalize several of the main Diophantine finiteness results in Chabauty-Kim theory to arbitrary number fields.

Boundedness of (ϵ,n)-complements for surfaces

We show Shokurov's complements conjecture holds for surfaces. More precisely, we show the existence of (ϵ,n)-complements for (ϵ,R)-complementary surface pairs when the coefficients of boundaries belong to a DCC set. For higher dimensional varieties, we show that Shokurov's complements conjecture implies the global index conjecture, Shokurov's index conjecture, and Shokurov's conjecture on Fano fibrations.

Divisor Varieties of Symmetric Products

Abstract The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and dimension) are smooth, irreducible projective varieties of known dimension. For higher dimensional varieties, the story is less well understood. Our purpose in this paper is to study in detail one class of higher dimensional examples where one can hope for a quite detailed picture, namely (the spaces parametrizing) divisors on the symmetric product of a curve.