Effective Divisors Research Articles

An algorithm for Hodge ideals

We present an algorithm to compute the Hodge ideals (see M. Mustaţă and M. Popa [Mem. Amer. Math. Soc. 262 (2019), pp. v + 80; J. Éc. polytech. Math. 6 (2019), pp. 283–328]) of Q \mathbb {Q} -divisors associated to any reduced effective divisor D D . The computation of the Hodge ideals is based on an algorithm to compute parts of the V V -filtration of Kashiwara and Malgrange on ι + O X ( ∗ D ) \iota _{+}\mathscr {O}_X(*D) and the characterization (see M. Mustaţă and M. Popa [Forum Math. Sigma 8 (2020), p. 41]) of the Hodge ideals in terms of this V V -filtration. In particular, this gives a new algorithm to compute the multiplier ideals and the jumping numbers of any effective divisor.

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.

On the intersection cohomology of the moduli of SLn$\mathrm{SL}_n$‐Higgs bundles on a curve

We explore the cohomological structure for the (possibly singular) moduli of SL n $\mathrm{SL}_n$ -Higgs bundles for arbitrary degree on a genus g $g$ curve with respect to an effective divisor of degree > 2 g − 2 $>2g-2$ . We prove a support theorem for the SL n $\mathrm{SL}_n$ -Hitchin fibration extending de Cataldo's support theorem in the nonsingular case, and a version of the Hausel–Thaddeus topological mirror symmetry conjecture for intersection cohomology. This implies a generalization of the Harder–Narasimhan theorem concerning semistable vector bundles for any degree. Our main tool is an Ngô–type support inequality established recently which works for possibly singular ambient spaces and intersection cohomology complexes.

Log Fano blowups of mixed products of projective spaces and their effective cones

We compute the cones of effective divisors on blowups of mathbb P^1 times mathbb P^2 and mathbb P^1 times mathbb P^3 in up to 6 points. We also show that all these varieties are log Fano, giving a conceptual explanation for the fact that all the cones we compute are rational polyhedral.

Quadrics and normal generation of line bundles on multiple coverings

In this work, a minimally FIIQ divisor on a smooth curve in is defined by an effective divisor which fails to impose independent conditions on quadrics in and all of whose proper subdivisors impose independent conditions on quadrics. Using this notion, we explicitly describe very ample non-special line bundles which fail to be normally generated on a simple -fold covering in terms of line bundles on under some numerical constraints. Furthermore, in case we obtain a concrete classification of such line bundles on

The Weak Gravity Conjecture and BPS Particles

AbstractMotivated by the Weak Gravity Conjecture, we uncover an intricate interplay between black holes, BPS particle counting, and Calabi‐Yau geometry in five dimensions. In particular, we point out that extremal BPS black holes exist only in certain directions in the charge lattice, and we argue that these directions fill out a cone that is dual to the cone of effective divisors of the Calabi‐Yau threefold. The tower and sublattice versions of the Weak Gravity Conjecture require an infinite tower of BPS particles in these directions, and therefore imply purely geometric conjectures requiring the existence of infinite towers of holomorphic curves in every direction within the dual of the cone of effective divisors. We verify these geometric conjectures in a number of examples by computing Gopakumar‐Vafa invariants.

The characterization of aCM line bundles on quintic hypersurfaces in $$\mathbb {P}^3$$

Let X be a smooth quintic hypersurface in \(\mathbb {P}^3\), let C be a smooth hyperplane section of X, and let \(H=\mathcal {O}_X(C)\). In this paper, we give a necessary and sufficient condition for the line bundle given by a non-zero effective divisor on X to be initialized and aCM with respect to H.

Uniformization of branched surfaces and Higgs bundles

Given a compact connected Riemann surface [Formula: see text] of genus [Formula: see text], and an effective divisor [Formula: see text] on [Formula: see text] with [Formula: see text], there is a unique cone metric on [Formula: see text] of constant negative curvature [Formula: see text] such that the cone angle at each point [Formula: see text] is [Formula: see text] [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on [Formula: see text] corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on [Formula: see text] parametrized by a nonempty open subset of [Formula: see text] that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin’s results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case [Formula: see text].

On the Mori theory and Newton–Okounkov bodies of Bott–Samelson varieties

We prove that on a Bott–Samelson variety X every movable divisor is nef. This enables us to consider Zariski decompositions of effective divisors, which in turn yields a description of the Mori chamber decomposition of the effective cone. This amounts to information on all possible birational morphisms from X. Applying this result, we prove the rational polyhedrality of the global Newton–Okounkov cone of a Bott–Samelson variety with respect to the so called ‘horizontal’ flag. In fact, we prove the stronger property of the finite generation of the corresponding global value semigroup.

Constructing tree decompositions of graphs with bounded gonality

In this paper, we give a constructive proof of the fact that the treewidth of a graph is at most its divisorial gonality. The proof gives a polynomial time algorithm to construct a tree decomposition of width at most k, when an effective divisor of degree k that reaches all vertices is given. We also give a similar result for two related notions: stable divisorial gonality and stable gonality.

Nevanlinna and algebraic hyperbolicity

Motivated by the notion of the algebraic hyperbolicity, we introduce the notion of Nevanlinna hyperbolicity for a pair [Formula: see text], where [Formula: see text] is a projective variety and [Formula: see text] is an effective Cartier divisor on [Formula: see text]. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard-type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by P[Formula: see text]un and Sibony [Value distribution theory for parabolic Riemann surfaces, preprint (2014), arXiv:1403.6596].

Gravitating vortices with positive curvature

We give a complete solution to the existence problem for gravitating vortices with non-negative topological constant c⩾0. Our first main result builds on previous results by Yang and establishes the existence of solutions to the Einstein-Bogomol'nyi equations, corresponding to c=0, in all admissible Kähler classes. Our second main result completely solves the existence problem for c>0. Both results are proved by the continuity method and require that a GIT stability condition for an effective divisor on the Riemann sphere is satisfied. For the former, the continuity path starts from a given solution with c=0 and deforms the Kähler class. For the latter result we start from the established solution in any fixed admissible Kähler class and deform the coupling constant α towards 0. A salient feature of our argument is a new bound Sg⩾c for the curvature of gravitating vortices, which we apply to construct a limiting solution along the path via Cheeger-Gromov theory.

Irreducible cone spherical metrics and stable extensions of two line bundles

A cone spherical metric is called irreducible if any developing map of the metric does not have monodromy in U(1). By using the theory of indigenous bundles, we construct on a compact Riemann surface X of genus gX≥1 a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible metrics with cone angles in 2πZ>1, which is generically injective in the algebro-geometric sense as gX≥2. As an application, we prove the following two results about irreducible metrics:• as gX≥2 and d is even and greater than 12gX−7, the effective divisors of degree d which could be represented by irreducible metrics form an arcwise connected Borel subset of Hausdorff dimension ≥2(d+3−3gX) in Symd(X);• as gX≥1, for almost every effective divisor D of degree odd and greater than 2gX−2 on X, there exist finitely many cone spherical metrics representing D.

On the Lipman–Zariski conjecture for logarithmic vector fields on log canonical pairs

We consider a version of the Lipman-Zariski conjecture for logarithmic vector fields and logarithmic $1$-forms on pairs. Let $(X,D)$ be a pair consisting of a normal complex variety $X$ and an effective Weil divisor $D$ such that the sheaf of logarithmic vector fields (or dually the sheaf of reflexive logarithmic $1$-forms) is locally free. We prove that in this case the following holds: If $(X,D)$ is dlt, then $X$ is necessarily smooth and $\lfloor D\rfloor $ is snc. If $(X,D)$ is lc or the logarithmic $1$-forms are locally generated by closed forms, then $(X,\lfloor D\rfloor)$ is toroidal.

On Syntomic Complexes with Modulus for Semi-stable Reduction Cases

In this paper, we define a syntomic complex for modulus pair $(X,D)$, where $X$ is a regular semi-stable family and $D$ is an effective Cartier divisor on $X$ and we compute its cohomology sheaves. We construct a symbol map for this syntomic complex for modulus pair $(X,D)$ and investigate its cokernel by computing its cohomology sheaves. To compute the structure of the cohomology sheaves of syntomic complex with modulus, we define some filtrations on it.

Analogues of Alladi's formula over global function fields

In this paper, we show an analogue of Kural, McDonald and Sah's result on Alladi's formula for global function fields. Explicitly, we show that for a global function field K, if a set S of prime divisors has a natural density δ(S) within prime divisors, then−limn→∞⁡∑1≤deg⁡D≤nD∈D(K,S)μ(D)|D|=δ(S), where μ(D) is the Möbius function on divisors and D(K,S) is the set of all effective distinguishable divisors whose smallest prime factors are in S. As applications, we get the analogue of Dawsey's and Sweeting and Woo's results to the Chebotarev Density Theorem for function fields, and the analogue of Alladi's result to the Prime Polynomial Theorem for arithmetic progressions. We also display a connection between the Möbius function and the Fourier coefficients of modular form associated to elliptic curves. The proof of our main theorem is similar to the approach in Kural et al.'s article.

Greatest common divisors of integral points of numerically equivalent divisors

We generalize the G.C.D. results of Corvaja--Zannier and Levin on $\mathbb G_m^n$ to more general settings. More specifically, we analyze the height of a closed subscheme of codimension at least $2$ inside an $n$-dimensional Cohen-Macaulay projective variety, and show that this height is small when evaluated at integral points with respect to a divisor $D$ when $D$ is a sum of $n+1$ effective divisors which are all numerically equivalent to some multiples of a fixed ample divisor. Our method is inspired by Silverman's G.C.D. estimate as an application of Vojta's conjecture, which is substituted by a more general version of Schmidt's subspace theorem of Ru--Vojta in our proof.

Generators of invariant linear system on tropical curves for finite isometry group

For a tropical curve $\Gamma$ and a finite subgroup $K$ of the isometry group of $\Gamma$, we prove, extending the work by Haase, Musiker and Yu ([6]), that the $K$-invariant part of the complete linear system associated to a $K$-invariant effective divisor on $\Gamma$ is finitely generated

Value distribution of derivatives in polynomial dynamics

AbstractFor every$m\in \mathbb {N}$, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in$\mathbb {C}\setminus \{0\}$under the$m$th order derivatives of the iterates of a polynomials$f\in \mathbb {C}[z]$of degree$d>1$towards the harmonic measure of the filled-in Julia set offwith pole at$\infty $. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number fieldkfor a sequence of effective divisors on$\mathbb {P}^1(\overline {k})$having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of$\mathbb {C}^2$has a given eigenvalue.

$k$-differentials on curves and rigid cycles in moduli space

For g\geq 2, j=1,\dots,g and n\geq g+j we exhibit infinitely many new rigid and extremal effective codimension j cycles in \overline{\mathcal{M}}_{g,n} , the Deligne-Mumford compactification of the moduli of n -pointed curves of genus g . The extremal cycles constructed correspond to the strata of quadratic differentials and projections of these strata under forgetful morphisms. We further show the same holds for k -differentials with k\geq 3 if the strata are irreducible. We compute the class of the divisors in the case of quadratic differentials which contain the first known examples of effective divisors on \overline{\mathcal{M}}_{g,n} with negative \psi_i coefficients.