Divisor Class Research Articles

On local divisor class groups of complete intersections

Samuel conjectured in 1961 that a (Noetherian) local complete intersection ring that is a UFD in codimension at most three is itself a UFD. It is said that Grothendieck invented local cohomology to prove this fact. Following the philosophy that a UFD is nothing else than a Krull domain (that is, a normal domain, in the Noetherian case) with trivial divisor class group, we take a closer look at the Samuel–Grothendieck Theorem and prove the following generalization: Let A be a local Cohen–Macaulay ring.(1)A is a normal domain if and only if A is a normal domain in codimension at most 1.(2)Suppose that A is a normal domain and a complete intersection. Then the divisor class group of A is a subgroup of the projective limit of the divisor class groups of the localizations Ap, where p runs through all prime ideals of height at most 3 in A. We use this fact to describe for an integral Noetherian locally complete intersection scheme X the gap between the groups of Weil and Cartier divisors, generalizing in this case the classical result that these two concepts coincide if X is locally a UFD.

A positive/tropical critical point theorem and mirror symmetry

Call a Laurent polynomial W ‘complete’ if its Newton polytope is full-dimensional with zero in its interior. Suppose W is a Laurent polynomial with coefficients in the positive part of the field of (generalised) Puiseaux series. Here a Puiseaux or generalised Puiseux series (with exponents in R) is called ‘positive’ if the coefficient of its leading term is in R>0. We show that W has a unique positive critical point pcrit, i.e. all of whose coordinates are positive, if and only if W is complete. For any complete, positive Laurent polynomial W in r variables we also obtain from its positive critical point pcrit a canonically associated ‘tropical critical point’ dcrit∈Rr by considering the valuations of the coordinates of pcrit. Moreover we give a finite recursive construction of dcrit in terms of a generalisation of the Newton polytope that we call the ‘Newton datum’ of W.We show that this result is compatible with a general form of mutation, so that it can be applied in a cluster varieties setting. We also show that our theorem carries over to the case where the exponents of monomials appearing in W are not integral but in R, even though W is then no longer Laurent.Finally, we describe applications to both algebraic and symplectic toric geometry inspired by mirror symmetry. On the one hand, in the algebraic context of a complete toric variety XΣ we apply our results to obtain for any divisor class [D] satisfying a certain integrality property, a canonical choice of torus-invariant representative. This generalises the standard toric boundary divisor of XΣ to divisor classes other than the anti-canonical class. On the other hand, our result generalises a result of [11] and relates to the construction of canonical non-displaceable Lagrangian tori for toric symplectic orbifolds using [13,37].

An Exploration of Degeneracy in Abelian Varieties of Fermat Type

The term degenerate is used to describe abelian varieties whose Hodge rings contain exceptional cycles – Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this article we examine degeneracy through these different but related lenses. We specialize to a family of abelian varieties of Fermat type, namely Jacobians of hyperelliptic curves of the form y 2 = x m − 1 . We prove that the Jacobian of the curve is degenerate whenever m is an odd, composite integer. We explore the various forms of degeneracy for several examples, each illustrating different phenomena that can occur.

Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi–Yau Three-Folds

We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The obtained formulae express cohomologies in terms of divisor class intersections, and are adapted to the decomposition of the effective cone into Zariski chambers. In particular, we show this occurs on generalised del Pezzo surfaces, toric surfaces, and K3 surfaces. In the second part we use these surface results to derive formulae for all line bundle cohomology on a simple class of elliptically fibered Calabi–Yau three-folds. Computing such quantities is a crucial step in deriving the massless spectrum in string compactifications.

On the rational cuspidal divisor class groups of Drinfeld modular curves X0(𝔭r)

Let [Formula: see text] be the rational cuspidal divisor class group of the Drinfeld modular curve [Formula: see text] for a prime power level [Formula: see text]. We relate the rational cuspidal divisors of degree 0 on [Formula: see text] with [Formula: see text]-quotients, where [Formula: see text] is the Drinfeld discriminant function. As a result, we are able to determine explicitly the structure of [Formula: see text] for arbitrary prime [Formula: see text] and [Formula: see text].

Quadratic forms and Genus Theory: A link with 2‐descent and an application to nontrivial specializations of ideal classes

AbstractGenus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well‐known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and have coefficients in any principal ideal domain (PID) . When , we show that the Genus Theory map is the quadratic form version of the 2‐descent map on a certain hyperelliptic curve. As an application, we make a contribution to a question of Agboola and Pappas regarding a specialization problem of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of nontrivial specializations has density 1.

The Picard index of a surface with torus action

AbstractWe consider normal rational projective surfaces with torus action and provide a formula for their Picard index, that means the index of the Picard group inside the divisor class group. As an application, we classify the log del Pezzo surfaces with torus action of Picard number one up to Picard index $$ 10\,000 $$ 10 000 .

Homogeneous star-operations on graded integral domains

Let Γ be a torsion-free cancellative monoid and let R = ⊕ α ∈ Γ R α be a graded integral domain. In this paper, we define the concept of homogeneous star-operations, which is a generalization of that of star-operations, and apply some properties of star-operations. Also, we define the concept of homogeneous divisor classes, which is the graded integral domain version of that of divisor classes, and study some equivalent conditions for the set of homogeneous divisor classes H D ( R ) of R to be a group.

Quantum K-theory of incidence varieties

We prove a conjecture of Buch and Mihalcea in the case of the incidence variety X=Fl(1,n-1;n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=\ extrm{Fl}\\hspace{0.55542pt}(1,n-1;n)$$\\end{document} and determine the structure of its (T-equivariant) quantum K-theory ring. Our results are an interplay between geometry and combinatorics. The geometric side concerns Gromov–Witten varieties of 3-pointed genus 0 stable maps to X with markings sent to Schubert varieties, while on the combinatorial side are formulas for the (equivariant) quantum K-theory ring of X. We prove that the Gromov–Witten variety is rationally connected when one of the defining Schubert varieties is a divisor and another is a point. This implies that the (equivariant) K-theoretic Gromov–Witten invariants defined by two Schubert classes and a Schubert divisor class can be computed in the ordinary (equivariant) K-theory ring of X. We derive a positive Chevalley formula for the equivariant quantum K-theory ring of X and a positive closed formula for Littlewood–Richardson coefficients in the non-equivariant quantum K-theory ring of X. The Littlewood–Richardson rule in turn implies that non-empty Gromov–Witten varieties given by Schubert varieties in general position have arithmetic genus 0.

The divisor class group of a discrete polymatroid

In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.

Heights and quantitative arithmetic on stacky curves

Abstract In this paper, we investigate the theory of heights in a family of stacky curves following recent work of Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. We count rational points having bounded ESZ-B height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We also show that when the Euler characteristic of stacky curves is non-positive, the ESZ-B height coming from the anti-canonical divisor class fails to have the Northcott property. We prove that a stacky version of a conjecture of Vojta is equivalent to the $abc$ -conjecture.

Torsionfreeness for divisor class groups of toric rings of integral polytopes

In the present paper, we give some sufficient conditions for Cl(k[P]) to be torsionfree, where Cl(k[P]) denote the divisor class group of the toric ring k[P] of an integral polytope P. We prove that Cl(k[P]) is torsionfree if P is compressed, and Cl(k[P]) is torsionfree if P is a (0,1)-polytope which has at most dim⁡P+2 facets. Moreover, we characterize the toric rings of (0,1)-polytopes in the case Cl(k[P])≅Z.

Nonvarying, affine and extremal geometry of strata of differentials

AbstractWe prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata of k-differentials of infinite area are affine varieties for all k. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in Teichmüller dynamics.

Effective divisors on projectivized Hodge bundles and modular Forms

AbstractWe construct vector‐valued modular forms on moduli spaces of curves and abelian varieties using effective divisors in projectivized Hodge bundles over moduli of curves. Cycle relations tell us the weight of these modular forms. In particular, we construct basic modular forms for genus 2 and 3. We also discuss modular forms on the moduli of hyperelliptic curves. In that case, the relative canonical bundle is a pull back of a line bundle on a ‐bundle over the moduli of hyperelliptic curves and we extend that line bundle to a compactification so that its push down is (close to) the Hodge bundle and use this to construct modular forms. In the Appendix, we use our method to calculate divisor classes in the dual projectivized k‐Hodge bundle determined by Gheorghita–Tarasca and by Korotkin–Sauvaget–Zograf.

Characterization of projective varieties beyond varieties of minimal degree and del Pezzo varieties

Varieties of minimal degree and del Pezzo varieties are basic objects in projective algebraic geometry. Those varieties have been characterized and classified for a long time in many aspects. Motivated by the question “which varieties are the most basic and simplest except the above two kinds of varieties in view of geometry and syzygies?”, we give an upper bound of the graded Betti numbers in the quadratic strand and characterize the extremal cases.The extremal varieties of dimension n, codimension e, and degree d are exactly characterized by the following two types:(i)Varieties with d=e+2, depthX=n, and Green-Lazarsfeld index a(X)=0,(ii)Arithmetically Cohen-Macaulay varieties with d=e+3. This is a generalization of G. Castelnuovo, G. Fano, and E. Park's results on the number of quadrics and an extension of the characterizations of varieties of minimal degree and del Pezzo varieties in view of linear syzygies of quadrics due to K. Han and S. Kwak ([6,8,30,16]).In addition, we show that every variety X that belongs to (i) or (ii) is always contained in a unique rational normal scroll Y as a divisor. Also, we describe the divisor class of X in Y.

A tree-based algorithm for the integration of monomials in the Chow ring of the moduli space of stable marked curves of genus zero

The Chow ring of the moduli space of stable marked curves is generated by Keel's divisor classes. The top graded part of this Chow ring is isomorphic to the integers, generated by the class of a single point. In this paper, we give an equivalent graphical characterization on the monomials in this Chow ring, as well as the characterization on the algebraic reduction on such monomials. Moreover, we provide an algorithm for computing the intersection degree of tuples of Keel's divisor classes — we call it the forest algorithm; the complexity of which is O(n3) in the worst case, where n refers to the number of marks on the stable curves in the ambient moduli space. Last but not least, we introduce three identities on multinomial coefficients which naturally came into play, showing that they are all equivalent to the correctness of the base case of the forest algorithm.

The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree

AbstractFor a positive integer N, let be the modular curve over and its Jacobian variety. We prove that the rational cuspidal subgroup of is equal to the rational cuspidal divisor class group of when for any prime p and any squarefree integer M. To achieve this, we show that all modular units on can be written as products of certain functions , which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on under a mild assumption.

The rational torsion subgroup of J0(N)

Let N be a positive integer and let J0(N) be the Jacobian variety of the modular curve X0(N). For any prime p≥5 whose square does not divide N, we prove that the p-primary subgroup of the rational torsion subgroup of J0(N) is equal to that of the rational cuspidal divisor class group of X0(N), which is explicitly computed in [33]. Also, we prove the same assertion holds for p=3 under the extra assumption that either N is not divisible by 3 or there is a prime divisor of N congruent to −1 modulo 3.

Chow classes of divisors on stacks of pointed hyperelliptic curves

We calculate the classes of the universal hyperelliptic Weierstrass divisor $\bar{{\mathcal H}}_{g,w}$ and the universal $g^1_2$ divisor $\bar{{\mathcal H}}_{g,g^1_2}$. Our results are expressed in terms of a basis for $Cl(\bar{{\mathcal H}}_{g,1})$ and $Cl(\bar{{\mathcal H}}_{g,2})$ computed by Scavia.

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.