Divisor Class Research Articles

Computing Riemann–Roch polynomials and classifying hyper-Kähler fourfolds

We prove that a hyper-Kähler fourfold satisfying a mild topological assumption is of K3 [ 2 ] ^{[2]} deformation type. This proves in particular a conjecture of O’Grady stating that hyper-Kähler fourfolds of K3 [ 2 ] ^{[2]} numerical type are of K3 [ 2 ] ^{[2]} deformation type. Our topological assumption concerns the existence of two integral degree-2 cohomology classes satisfying certain numerical intersection conditions. There are two main ingredients in the proof. We first prove a topological version of the statement, by showing that our topological assumption forces the Betti numbers, the Fujiki constant, and the Huybrechts–Riemann–Roch polynomial of the hyper-Kähler fourfold to be the same as those of K3 [ 2 ] ^{[2]} hyper-Kähler fourfolds. The key part of the article is then to prove the hyper-Kähler SYZ conjecture for hyper-Kähler fourfolds for divisor classes satisfying the numerical condition mentioned above.

Some Properties of n-semidualizing Modules

Let R be a commutative noetherian ring. The n-semidualizing modules of R are generalizations of its semidualizing modules. We will prove some basic properties of n-semidualizing modules. Our main result and example shows that the divisor class group of a Gorenstein determinantal ring over a field is the set of isomorphism classes of its 1-semidualizing modules. Finally, we pose some questions about n-semidualizing modules.

Complete curves in the strata of differentials

Gendron proved that the strata of holomorphic differentials with prescribed orders of zeros do not contain complete algebraic curves by applying the maximum modulus principle to saddle connections. Here we provide an alternative proof for this result by using positivity of divisor classes on moduli spaces of curves.

Fractons on graphs and complexity

We introduce two exotic lattice models on a general spatial graph. The first one is a matter theory of a compact Lifshitz scalar field, while the second one is a certain rank-2 $U(1)$ gauge theory of fractons. Both lattice models are defined via the discrete Laplacian operator on a general graph. We unveil an intriguing correspondence between the physical observables of these lattice models and graph theory quantities. For instance, the ground state degeneracy of the matter theory equals the number of spanning trees of the spatial graph, which is a common measure of complexity in graph theory ("GSD = complexity"). The discrete global symmetry is identified as the Jacobian group of the graph. In the gauge theory, superselection sectors of fractons are in one-to-one correspondence with the divisor classes in graph theory. In particular, under mild assumptions on the spatial graph, the fracton immobility is proven using a graph-theoretic Abel-Jacobi map.

Orbits of the Automorphism Group of Horospherical Varieties, and Divisor Class Group

В 2013 г. Бажов доказал критерий того, что две точки полного торического многообразия лежат в одной орбите связной компоненты единицы группы автоморфизмов. Этот критерий сформулирован в терминах группы классов дивизоров. В том же году Аржанцев и Бажов получили аналогичный критерий для аффинного торического многообразия. В работе доказывается необходимое условие, аналогичное этому критерию, в случаях аффинного и проективного орисферических многообразий.

Orbits of the Automorphism Group of Horospherical Varieties, and Divisor Class Group

Orbits of the Automorphism Group of Horospherical Varieties, and Divisor Class Group

Effective divisor classes on metric graphs

We introduce the notion of semibreak divisors on metric graphs (tropical curves) and prove that every effective divisor class (of degree at most the genus) has a semibreak divisor representative. This appropriately generalizes the notion of break divisors (in degree equal to genus). Our method of proof is new, even for the special case of break divisors. We provide an algorithm to efficiently compute such semibreak representatives. Semibreak divisors provide the tool to establish some basic properties of effective loci inside Picard groups of metric graphs. We prove that effective loci are pure-dimensional polyhedral sets. We also prove that a `generic' divisor class (in degree at most the genus) has rank zero, and that the Abel-Jacobi map is `birational' onto its image. These are analogues of classical results for Riemann surfaces.

Surface bundles and the section conjecture

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of this conjecture. In so doing we produce many examples of curves satisfying the section conjecture over fields of geometric interest, and then over p-adic fields and number fields via a Chebotarev argument. We construct two Galois cohomology classes o_1 and o_2, which obstruct the existence of pi_1-sections and hence of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli space of curves M_g studied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes obstruct sections. We then use these constructions to produce curves over p-adic fields and number fields where each class obstructs pi_1-sections and hence rational points. Among our geometric results are a new proof of the section conjecture for the generic curve of genus g>2, and a proof of the section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the existence of a section is genuinely non-abelian).

Intersection theory of nefb-divisor classes

We prove that any nef$b$-divisor class on a projective variety defined over an algebraically closed field of characteristic zero is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef$b$-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint-free curve class is a power of a nef$b$-divisor, and relate this statement to the Zariski decomposition of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces contained in the space of$b$-divisors which were defined in our previous work.

A Lefschetz hyperplane theorem for non-Archimedean Jacobians

We establish a Lefschetz hyperplane theorem for the Berkovich analytifications of Jacobians of curves over an algebraically closed non-Archimedean field. Let J be the Jacobian of a curve X, and let $$W_{d} \subset J$$ be the locus of effective divisor classes of degree d. We show that the pair $$(J^{an},W_d^{an})$$ is d-connected, and thus in particular the inclusion of the analytification of the theta divisor $$\Theta ^{an}$$ into $$J^{an}$$ satisfies a Lefschetz hyperplane theorem for $${\mathbb {Z}}$$ -cohomology groups and homotopy groups. A key ingredient in our proof is a generalization, over arbitrary characteristics and allowing arbitrary singularities on the base, of a result of Brown and Foster for the homotopy type of analytic projective bundles.

COMBINED COUNT OF REAL RATIONAL CURVES OF CANONICAL DEGREE 2 ON REAL DEL PEZZO SURFACES WITH

AbstractWe propose two systems of “intrinsic” weights for counting such curves. In both cases the result acquires an exceptionally strong invariance property: it does not depend on the choice of a surface. One of our counts includes all divisor classes of canonical degree 2 and gives in total 30. The other one excludes the class $-2K$ , but adds up the results of counting for a pair of real structures that differ by Bertini involution. This count gives 96.

Class groups of open Richardson varieties in the Grassmannian are trivial

We prove that the divisor class group of any open Richardson variety in the Grassmannian is trivial. Our proof uses Nagata’s criterion, localizing the coordinate ring at a suitable set of Plücker coordinates. We prove that these Plücker coordinates are prime elements by showing that the subscheme they define is an open subscheme of a positroid variety. Our results hold over any field and over the integers.

The rational cuspidal divisor class group of X0(N)

For any positive integer N, we completely determine the structure of the rational cuspidal divisor class group of X0(N), which is conjecturally equal to the rational torsion subgroup of J0(N). More specifically, for a given prime ℓ, we construct a rational cuspidal divisor Zℓ(d) for any non-trivial divisor d of N. Also, we compute the order of the linear equivalence class of Zℓ(d) and show that the ℓ-primary subgroup of the rational cuspidal divisor class group of X0(N) is isomorphic to the direct sum of the cyclic subgroups generated by the linear equivalence classes of Zℓ(d).

Reduction of Divisors and the Clebsch System

There are a few Lax matrices of the Clebsch system. Poles of the Baker – Akhiezer function determine the class of equivalent divisors on the corresponding spectral curves. According to the Riemann – Roch theorem, each class has a unique reduced representative. We discuss properties of such a reduced divisor on the spectral curve of $$3\times 3$$ Lax matrix having a natural generalization to $$gl^{*}(n)$$ case.

On the distribution of prime divisors in Krull monoid algebras

In the present work, we prove that every class of the divisor class group of a Krull monoid algebra contains infinitely many prime divisors. Several attempts to this result have been made in the literature so far, unfortunately with open gaps. We present a complete proof of this fact.

The Divisors of Prym Semicanonical Pencils

Abstract In the moduli space ${{\mathcal {R}}}_g$ of double étale covers of curves of a fixed genus $g$, the locus of covers of curves with a semicanonical pencil decomposes as the union of two divisors—${{\mathcal {T}}}^e_g$ and ${{\mathcal {T}}}^o_g$. Adapting arguments of Teixidor for the divisor of curves having a semicanonical pencil, we prove that both divisors are irreducible and compute their divisor classes in the Deligne–Mumford compactification ${\overline {{\mathcal {R}}}}_g$.

Borcherds’ Method for Enriques Surfaces

We classify all primitive embeddings of the lattice of numerical equivalence classes of divisors of an Enriques surface with the intersection form multiplied by 2 into an even unimodular hyperbolic lattice of rank 26. These embeddings have a property that facilitates the computation of the automorphism group of an Enriques surface by Borcherds’ method.

Canonical Modules and Class Groups of Rees-Like Algebras

Rees-like algebras have played a major role in settling the Eisenbud–Goto conjecture. This paper concerns the structure of the canonical module of the Rees-like algebra and its class groups. Via an explicit computation based on linkage, we provide an explicit and surprisingly well-structured resolution of the canonical module in terms of a type of double-Koszul complex. Additionally, we give descriptions of both the divisor class group and the Picard group of a Rees-like algebra.

On the Lefschetz standard conjecture for Lagrangian covered hyper-Kähler varieties

We investigate the Lefschetz standard conjecture for degree 2 cohomology of hyper-Kähler manifolds admitting a covering by Lagrangian subvarieties. In the case of a Lagrangian fibration, we show that the Lefschetz standard conjecture is implied by the SYZ conjecture characterizing classes of divisors associated with Lagrangian fibration. In dimension 4, we consider the more general case of a Lagrangian covered fourfold X, and prove the Lefschetz standard conjecture in degree 2, assuming ρ(X)=1 and X is general in moduli. Finally we discuss various links between Lefschetz cycles and the study of the rational equivalence of points and Bloch-Beilinson type filtrations, giving a general interpretation of a recent intriguing result of Marian and Zhao.

Dynamical invariants and intersection theory on the flex and gothic loci

The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points. The flex and gothic loci provide the first example of a primitive, totally geodesic subvariety of moduli space and new \({\text {SL}}_2({\mathbb {R}})\)-invariant varieties in Teichmüller dynamics, as discovered by McMullen–Mukamel–Wright. In this paper we determine the divisor class of the flex locus as well as various tautological intersection numbers on the gothic locus. For the case of the gothic locus our result confirms numerically a conjecture of Chen–Möller–Sauvaget about computing sums of Lyapunov exponents for \({\text {SL}}_2({\mathbb {R}})\)-invariant varieties via intersection theory.