Boundary Divisors Research Articles

The class of the locus of intermediate Jacobians of cubic threefolds

We study the locus of intermediate Jacobians of cubic threefolds within the moduli space of complex principally polarized abelian fivefolds, and its generalization to arbitrary genus - the locus of abelian varieties with a singular odd two-torsion point on the theta divisor. Assuming that this locus has expected codimension (which we show to be true for genus up to 5), we compute the class of this locus, and of is closure in the perfect cone toroidal compactification, in the Chow, homology, and the tautological ring. We work out the cases of genus up to 5 in detail, obtaining explicit expressions for the classes of the closures of the locus of products of an elliptic curve and a hyperelliptic genus 3 curve, in moduli of principally polarized abelian fourfolds, and of the locus of intermediate Jacobians in genus 5. In the course of our computation we also deal with various intersections of boundary divisors of a level toroidal compactification, which is of independent interest in understanding the cohomology and Chow rings of the moduli spaces.

Fulton's conjecture for M̅ 0,7

Fulton's conjecture for the moduli space of stable pointed rational curves, \bar{M}_{0,n}, claims that a divisor non-negatively intersecting all F-curves is linearly equivalent to an effective sum of boundary divisors. Our main result is a proof of Fulton's conjecture for n=7. A key ingredient in the proof is an \binom{n}{4} dimensional-subspace of the Neron-Severi space of \bar{M}_{0,n}, defined by averages of Keel relations, for which we prove Fulton's conjecture for all n.

A direct proof of BCFW recursion for twistor-strings

This paper gives a direct proof that the leading trace part of the genus zero twistor-string path integral obeys the BCFW recursion relation. This is the first complete proof that the twistor-string correctly computes all tree amplitudes in maximally supersymmetric Yang-Mills theory. The recursion has a beautiful geometric interpretation in twistor space that closely reects the structure of BCFW recursion in momentum space, both on the one hand as a relation purely among tree amplitudes with shifted external momenta, and on the other as a relation between tree amplitudes and leading singularities of higher loop amplitudes. The proof works purely at the level of the string path integral and is intimately related to the recursive structure of boundary divisors in the moduli space of stable maps to $ \mathbb{C}{\mathbb{P}^3} $ .

Families over special base manifolds and a conjecture of Campana

Consider a smooth, projective family of canonically polarized varieties over a smooth, quasi-projective base manifold Y, all defined over the complex numbers. It has been conjectured that the family is necessarily isotrivial if Y is special in the sense of Campana. We prove the conjecture when Y is a surface or threefold. The proof uses sheaves of symmetric differentials associated to fractional boundary divisors on log canonical spaces, as introduced by Campana in his theory of Orbifoldes Géométriques. We discuss a weak variant of the Harder–Narasimhan Filtration and prove a version of the Bogomolov–Sommese Vanishing Theorem that take the additional fractional positivity along the boundary into account. A brief, but self-contained introduction to Campana’s theory is included for the reader’s convenience.

Swan conductors for p-adic differential modules. II Global variation

AbstractUsing a local construction from a previous paper, we exhibit a numerical invariant, the differential Swan conductor, for an isocrystal on a variety over a perfect field of positive characteristic overconvergent along a boundary divisor; this leads to an analogous construction for certain p-adic and l-adic representations of the étale fundamental group of a variety. We then demonstrate some variational properties of this definition for overconvergent isocrystals, paying special attention to the case of surfaces.

On asymptotics of complete Ricci-flat Kähler metrics on open manifolds

Tian and Yau constructed in J. Am. Math. Soc., 3(3):579–609, 1990, a complete Ricci-flat Kahler metric on the complement of an ample and smooth anticanonical divisor. For the explicitly constructed referential metric ω of Tian and Yau (J. Am. Math. Soc., 3(3):579–609, 1990) we prove a property that \({\|\partial\overline\partial u\|_\omega}\) has the same decay rate as Δωu provided u satisfies some decay conditions on higher Laplacians. As an application we describe the behaviour of this metric towards the boundary divisor and prove the best possible decay rate of the difference to ω.

Sur la première classe de Stiefel-Whitney de l’espace des applications stables réelles vers l’espace projectif

Moduli space of genus zero stable maps to the projective three-space naturally carries a real structure such that the fixed locus is a moduli space for real rational spatial curves with real marked points. The latter is a normal projective real variety. The singular locus being in codimension at least two, a first Stiefel-Whitney class is well defined. In this paper, we determine a representative for the first Stiefel-Whitney class of such real space when the evaluation map is generically finite. This can be done by means of Poincaré duals of boundary divisors.

Limits of families of Brieskorn lattices and compactified classifying spaces

We investigate variations of Brieskorn lattices over non-compact parameter spaces, and discuss the corresponding limit objects on the boundary divisor. We study the associated variation of twistors and the corresponding limit mixed twistor structures. We construct a compact classifying space for regular singular Brieskorn lattices and prove that its pure polarized part carries a natural hermitian structure and that the induced distance makes it into a complete metric space.

Some intersection numbers of divisors on toroidal compactifications of 𝒜_{ℊ}

We study the top intersection numbers of the boundary and Hodge class divisors on toroidal compactifications of the moduli space A g \mathcal {A}_g of principally polarized abelian varieties and compute those numbers that live away from the stratum which lies over the closure of A g − 3 \mathcal {A}_{g-3} in the Satake compactification.

The Cox ring of $\overline {M}_{0,6}$

We prove that the Cox ring of the moduli space V 0,6 , of stable rational curves with 6 marked points, is finitely generated by sections corresponding to the boundary divisors and divisors which are pull-backs of the hyperelliptic locus in M 3 via morphisms p: M 0,6 → M 3 that send a 6-pointed rational curve to a curve with 3 nodes by identifying 3 pairs of points. In particular this gives a self-contained proof of Hassett and Tschinkel's result about the effective cone of M 0,6 being generated by the above mentioned divisors.

Première classe de Stiefel–Whitney des espaces d'applications stables réelles en genre zéro vers une surface convexe

RésuméSoit (X, cX) une surface projective convexe munie d'une structure réelle. L'espace de module des applications stables $\smash{\overline{\mathcal{M}}_{0,k}(X,d)}$ peut être équipé de plusieurs structures réelles associées à cX et à un élément d'ordre deux τ du groupe des permutations Sk qui agit sur les points marqués. La partie réelle correspondant à chaque structure ℝτ$\smash{\overline{\mathcal{M}}_{0,k}(X,d)}$ est une variété réelle projective et normale. Puisque le lieu singulier est de codimension au moins deux, ces espaces possèdent une première classe de Stiefel–Whitney pour laquelle on détermine un représentant dans le cas k = c1(X)d − 1 où c1(X) désigne la première classe de Chern du fibré tangent de X. Plus précisément, nous donnons une description homologique des classes de Stiefel–Whitney en termes de partie réelle de certains diviseurs de la frontière de l'espace des applications stables.

The Effective Cone of the Kontsevich Moduli Space

AbstractIn this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, , stabilize when r ≥ d. We give a complete characterization of the effective divisors on . They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.

Positive divisors on quotients of [formula omitted] and the Mori cone of [formula omitted

We prove that if m ≥ n − 3 then every S m -invariant F -nef divisor on the moduli space of stable n -pointed curves of genus zero is linearly equivalent to an effective combination of boundary divisors. As an application, we determine the Mori cone of the moduli spaces of stable curves of small genus with few marked points.

A Presentation for the Chow Ring of

We give a presentation for the Chow ring of the moduli space of degree 2 stable maps from 2-pointed rational curves to the projective line. Also, integrals of all degree 4 monomials in the hyperplane pullbacks and boundary divisors of this ring are computed using equivariant intersection theory. Finally, the presentation is used to give a new computation of the (previously known) values of the genus zero, degree 2, 2-pointed gravitational correlators of the projective line.

Compactifications of subvarieties of tori

We study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non-archimedean amoebas. These compactifications have many nice properties, for example any k boundary divisors intersect in codimension k. We consider some examples including M0,n ⊂ M0,n (and more generally log canonical models of complements of hyperplane arrangements) and compact quotients of Grassmannians by a maximal torus.

Affine threefolds whose log canonical bundles are not numerically effective

Let X ↪ ( T , D ) be a compactification of an affine 3-fold X into a smooth projective 3-fold T such that the (reduced) boundary divisor D is SNC. In this paper, as an affine counterpart to the work due to S. Mori (cf. [S. Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982) 133–176]), we shall classify ( K + D ) -negative extremal rays on T . In particular, if such an extremal ray R = R + [ C ] intersects K non-negatively, we shall describe the log flips and divisorial contractions appearing explicitly.

Toroidal crossings and logarithmic structures

We generalize Friedman's notion of d-semistability, which is a necessary condition for spaces with normal crossings to admit smoothings with regular total space. Our generalization deals with spaces that locally look like the boundary divisor in Gorenstein toroidal embeddings. In this situation, we replace d-semistability by the existence of global log structures for a given gerbe of local log structures. This leads to cohomological descriptions for the obstructions, existence, and automorphisms of log structures. We also apply toroidal crossings to mirror symmetry, by giving a duality construction involving toroidal crossing varieties whose irreducible components are toric varieties. This duality reproduces a version of Batyrev's construction of mirror pairs for hypersurfaces in toric varieties, but it applies to a larger class, including degenerate abelian varieties.

Equivariant vector bundles on group completions

In this paper, we describe the category of bi-equivariant vector bundles on a bi-equivariant smooth (partial) compactification of a connected reductive algebraic group with normal crossing boundary divisors. Our result is a generalization of the description of the category of equivariant vector bundles on toric varieties established by A. A. Klyachko (Math. USSR Izvest. 35 No. 2 (1990)). As an application, we prove splitting of equivariant vector bundles of low rank on the wonderful compactification of an adjoint simple group in the sense of C. De Concini and C. Procesi (Lect. Notes Math. 996 (1983)). Moreover, we present an answer to a problem raised by B. Kostant in the case of complex groups.

Algebraic torus actions on affine algebraic surfaces

We investigate smooth affine surfaces with G m -actions and A ∗ 1 -fibrations. We consider a smooth affine surface with an untwisted A ∗ 1 -fibration over the affine line, and obtain a characterization of the fibration is unique in terms of the boundary divisor of the surface. As an application of the characterization, we know that a smooth affine surface admitting two independent G m -actions is isomorphic to A ∗ 1 × A ∗ 1 or A 1 × A ∗ 1 unless S has an A 1 -fibration over A 1 .

A logarithmic view towards semistable reduction

A smooth, proper family of curves creates a monodromy action of the fundamental group of the base on the H of a fibre. The geometric condition of T. Saito for the action of the wild inertia of a boundary point to be trivial is transformed to the condition of logarithmic smooth reduction. The proof emphasizes methods and results from logarithmic geometry. It applies to quasi–projective smooth curves with etale boundary divisor.