Enumerative Geometry Research Articles

Tautological module and intersection theory on Hilbert schemes of nodal curves

This paper presents the rudiments of Hilbert-Mumford Intersection (HMI) theory: intersection theory on the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension. We introduce an additive group of geometric cycles, called ’tautological module’, generated by diagonal loci, node scrolls, and twists thereof. We determine recursively the intersection action on this group by the discriminant (big diagonal) divisor and all its powers. We show that this suffices to determine arbitrary polynomials in Chern classes, in particular Chern numbers, for the tautological vector bundles on the Hilbert schemes, which are closely related to enumerative geometry of families of nodal curves.

Geometry of Spectral Curves and All Order Dispersive Integrable System

We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large. The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in 1/N an isomonodromic problem given by a Lax pair, and the relation between "correlators", the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.

Thom series of contact singularities

Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this paper we develop a general method for calculating Thom polynomials of contact singularities. Along the way, relations with the equivariant geometry of (punctual, local) Hilbert schemes, and with iterated residue identities are revealed.

Equivariant classes of matrix matroid varieties

To each subset I of \{1,\dots,k\} associate an integer r(I) . Denote by X the collection of those n\times k matrices for which the rank of a union of columns corresponding to a subset I is r(I) , for all I . We study the equivariant cohomology class represented by the Zariski closure Y=\overline{X} . This class is an invariant of the underlying matroid structure. Its calculation incorporates challenges similar to the calculation of the ideal of Y , namely, the determination of the geometric theorems for the matroid. This class also gives information on the degenerations and hierarchy of matroids. New developments in the theory of Thom polynomials of contact singularities (namely, a recently found stability property) help us to calculate these classes and present their basic properties. We also show that the coefficients of this class are solutions to problems in enumerative geometry, which are natural generalization of the linear Gromov–Witten invariants of projective spaces.

A Fourier‐Mukai approach to the enumerative geometry of principally polarized abelian surfaces

AbstractWe study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$({\mathbb T},\ell )$\end{document}. Using Fourier‐Mukai techniques we associate certain jumping schemes to such sheaves and completely classify such loci. We give examples of applications to the enumerative geometry of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb T}$\end{document} and show that no smooth genus 5 curve on such a surface can contain a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$g^1_3$\end{document}. We also describe explicitly the singular divisors in the linear system |2ℓ|.

Lagrangian topology and enumerative geometry

We use the "pearl" machinery in our previous work to study certain enumerative invariants associated to monotone Lagrangian submanifolds.

Landau-Ginzburg models in real mirror symmetry

Récemment, la symétrie miroir pour les cordes ouvertes a dévoilé de nouveaux liens entre la géométrie symplectique et énumérative (modèle A) et la géométrie algébrique complexe (modèle B) qui en un certain sens se situent entre la symétrie miroir classique et sa version homologique. On résume ici le rôle que jouent dans cette histoire les factorisations matricielles et la correspondance Calabi-Yau/Landau-Ginzburg.

The genus zero Gromov–Witten invariants of the symmetric square of the plane

We study the Abramovich–Vistoli moduli space of genus zero orbifold stable maps to [Sym P], the stack symmetric square of P. This space compactifies the moduli space of stable maps from hyperelliptic curves to P, and we show that all genus zero Gromov–Witten invariants are determined from trivial enumerative geometry of hyperelliptic curves. We also show how the genus zero Gromov–Witten invariants can be used to determine the number of hyperelliptic curves of degree d and genus g interpolating 3d+ 1 generic points in P. Comparing our method to that of Graber for calculating the same numbers, we verify an example of the crepant resolution conjecture.

Algebraic combinatorics of diametric magic circles

Abstract We provide a definition of a diametric magic circle of order n. In this paper, we use techniques in computational algebraic combinatorics and enumerative geometry to construct and to count d-magic circles. We also provide a description of its minimal Hilbert basis and determine the symmetry operations on d-magic circles. Finally, we give an algorithm for writing a natural d-magic circle in terms of the minimal Hilbert basis.

Noncommutative Schur polynomials and the crystal limit of the -vertex model

Starting from the Verma module of we consider the evaluation module for affine and discuss its crystal limit (q → 0). There exists an associated integrable statistical mechanics model on a square lattice defined in terms of vertex configurations. Its transfer matrix is the generating function for noncommutative complete symmetric polynomials in the generators of the affine plactic algebra, an extension of the finite plactic algebra first discussed by Lascoux and Schützenberger. The corresponding noncommutative elementary symmetric polynomials were recently shown to be generated by the transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin and Kitanine. Here we establish that both generating functions satisfy Baxter's TQ-equation in the crystal limit by tying them to special solutions of the Yang–Baxter equation. The TQ-equation amounts to the well-known Jacobi–Trudi formula leading naturally to the definition of noncommutative Schur polynomials. The latter can be employed to define a ring which has applications in conformal field theory and enumerative geometry: it is isomorphic to the fusion ring of the Wess–Zumino–Novikov–Witten model whose structure constants are the dimensions of spaces of generalized θ-functions over the Riemann sphere with three punctures.

The general quadruple point formula

Maps between manifolds $M^m\to N^{m+\ell}$ ($\ell>0$) have multiple points, and more generally, multisingularities. The closure of the set of points where the map has a particular multisingularity is called the multisingularity locus. There are universal relations among the cohomology classes represented by multisingularity loci, and the characteristic classes of the manifolds. These relations include the celebrated Thom polynomials of monosingularities. For multisingularities, however, only the form of these relations is clear in general (due to Kazarian), the concrete polynomials occurring in the relations are much less known. In the present paper we prove the first general such relation outside the region of Morin-maps: the general quadruple point formula. We apply this formula in enumerative geometry by computing the number of 4-secant linear spaces to smooth projective varieties. Some other multisingularity formulas are also studied, namely 5, 6, 7 tuple point formulas, and one corresponding to $\Sigma^2\Sigma^0$ multisingularities.

Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations

We compute the recently introduced Fan-Jarvis-Ruan-Witten theory of W-curves in genus zero for quintic polynomials in five variables and we show that it matches the Gromov-Witten genus-zero theory of the quintic three-fold via a symplectic transformation. More specifically, we show that the J-function encoding the Fan-Jarvis-Ruan-Witten theory on the A-side equals via a mirror map the I-function embodying the period integrals at the Gepner point on the B-side. This identification inscribes the physical Landau-Ginzburg/Calabi-Yau correspondence within the enumerative geometry of moduli of curves, matches the genus-zero invariants computed by the physicists Huang, Klemm, and Quackenbush at the Gepner point, and yields via Givental's quantization a prediction on the relation between the full higher genus potential of the quintic three-fold and that of Fan-Jarvis-Ruan-Witten theory.

The Kodaira dimension of the moduli space of Prym varieties

We study the enumerative geometry of the moduli space \mathcal R_g of Prym varieties of dimension g–1 . Our main result is that the compactication of \mathcal R_g is of general type as soon as g > 13 and g is different from 15 . We achieve this by computing the class of two types of cycles on \mathcal R_g : one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym–Green conjecture on syzygies of Prym-canonical curves. We also perform a detailed study of the singularities of the Prym moduli space, and show that for g ≥4 , pluricanonical forms extend to any desingularization of the moduli space.

Enumerative geometry of divisorial families of rational curves

Abstract. We compute the number of irreducible rational curves of given degree with 1 tacnode in P2 or 1 node in P3 meeting an appropriate generic collection of points and lines. As a byproduct, we also compute the number of rational plane curves of degree d passing through 3d − 2 given points and tangent to a given line. The method is ‘classical’, free of Quantum Cohomology.

On the enumerative geometry of the resultant variety

Using the admissibility condition of Scheja and Storch and/or the polynomial condition of Friedlander and Halperin a complete description of all maximal linear subspaces of the resultant variety in the isotropic and the anisotropic case is given.

Komplexe Analysis

The workshop Komplexe Analysis , organised by Jean-Pierre Demailly (Grenoble), Klaus Hulek (Hannover), Ngaiming Mok (Hong Kong) and Thomas Peternell (Bayreuth) was held August 24th–August 30, 2008. This meeting was well attended with 46 participants from Europe, US, and the Far East. The participants included several leaders in the field as well as many young (non-tenured) researchers. The aim of the meeting was to present recent important results in several complex variables and complex geometry with particular emphasis on topics linking different areas of the field, as well as to discuss new directions and open problems. Altogether there were nineteen talks of 60 minutes each, a programme which left sufficient time for informal discussions and joint work on research projects. One of the topics at the center of the conference was the classification theory of higher dimensional varieties. Y. Kawamata lectured on the connections between the minimal model programme and derived categories; A. Corti discussed an approach to the finite generation of the canonical ring without minimal models, but still in connection with the seminal work which was presented by J. McKernan in the last Complex Analysis meeting in Oberwolfach 2006, where the finite generation of the canonical ring of varieties of general type was announced. Extension theorems, non vanishing and positivity result for certain direct image sheaves play a role in the global classification of complex manifolds. This was largely discussed by M. Paun and B. Berndtsson. In their work analytic methods are central, whereas the talks by Kawamata and Corti were more of an algebraic nature. Also very much on the analytic side and connected to Berndtsson's talk, H. Tsuji lectured on generalised Kähler–Einstein metrics. Families of projective manifolds over higher-dimensional base spaces were considered in the talk by S. Kebekus. Direct images of coherent sheaves also play a central role in this context. About five years ago, Campana introduced new variations on the concept of “orbifolds”; they were already the suject of talks in past sessions and have turned out to be of increasing interest – in the present session, new results on the hyperbolicity of orbifolds were presented in the talk by E. Rousseau. As to varieties with special geometry, K. Oguiso spoke on non-algebraic hyperkähler manifolds and, with a rather different flavour, F. Catanese on complex and real threefolds fibered by rational curves, with a special emphasis on real algebraic geometry. J. Chen discussed the influence of terminal singularities in three-dimensional geometry, a more algebraic topic. On the analytic side, A. Teleman reported on recent progress in the classification of non-Kähler surfaces in the so called Kodaira class VII, using gauge-theoretical methods, and S. K. Yeung lectured on new results on fake projective planes. Group actions and envelopes of holomorphy were the topics of the talk by X. Zhou. S. Boucksom discussed equidistribution of Fekete points on complex manifolds, in relation with energy functionals for Monge-Ampère operators. R. Lazarsfeld presented a very interesting new approach to study properties of linear systems and line bundles via convex geometry. Overall, moduli spaces appeared to be a central theme in the workshop, and were discussed extensively in at least four talks: V. Gritsenko considered moduli spaces of K3-surfaces; S. Grushevsky spoke on intersection numbers of divisor on the moduli space of curves, and K. Ludwig and G. Farkas lectured on the moduli spaces of spin and Prym curves, their singularities, Kodaira dimension and enumerative geometry.

The 3–fold vertex via stable pairs

The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3-folds. We evaluate the equivariant vertex for stable pairs on toric 3-folds in terms of weighted box counting. In the toric Calabi-Yau case, the result simplifies to a new form of pure box counting. The conjectural equivalence with the DT vertex predicts remarkable identities. The equivariant vertex governs primary insertions in the theory of stable pairs for toric varieties. We consider also the descendent vertex and conjecture the complete rationality of the descendent theory for stable pairs.

First steps in tropical intersection theory

We establish first parts of a tropical intersection theory. Namely, we define cycles, Cartier divisors and intersection products between these two (without passing to rational equivalence) and discuss push-forward and pull-back. We do this first for fans in $${\mathbb{R}^{n}}$$ and then for “abstract” cycles that are fans locally. With regard to applications in enumerative geometry, we finally have a look at rational equivalence and intersection products of cycles and cycle classes in $${\mathbb{R}^{n}}$$ .

Divisors in the moduli spaces of curves

The calculation by Harer [12] of the second homology groups of the moduli spaces of smooth curves over C can be regarded as a major step towards the understanding of the enumerative geometry of the moduli spaces of curves [21, 17]. However, from the point of view of an algebraic geometer, Harer’s approach has the drawback of being entirely transcendental; in addition, his proof is anything but simple. It would be desirable to provide a proof of his result which is more elementary, and algebro-geometric in nature. While this cannot be done at the moment, as we shall explain in this note it is possible to reduce the transcendental part of the proof, at least for homology with rational coefficients, to a single result, also due to Harer [13], asserting that the homology of Mg,n, the moduli space of smooth n-pointed genus g curves, vanishes above a certain explicit degree. A sketch of the proof of Harer’s vanishing theorem, which is not at all difficult, will be presented in Section 5 of this survey. It must be observed that Harer’s vanishing result is an immediate consequence of an attractive algebro-geometric conjecture of Looijenga (Conjecture 1 in Section 5); an affirmative answer to the conjecture would thus give a completely algebro-geometric proof of Harer’s theorem on the second rational homology of moduli spaces of curves. In this note we describe how one can calculate the first and second rational (co)homology groups of Mg,n, and those of Mg,n, the moduli space of stable n-pointed curves of genus g, using only relatively simple algebraic geometry and Harer’s vanishing theorem. ForMg,n, this program was carried out in [5], where the third and fifth cohomology groups were also calculated and shown to always vanish; in Section 6, we give an outline of the argument, which uses in an essential way a simple Hodge-theoretic result due to Deligne [10]. In genus zero, we rely on Keel’s calculation of the Chow ring of M0,n; a simple proof of Keel’s result in the case of divisors is presented in Section 4. We finally give a new proof of Harer’s theorem for H(Mg,n;Q); we also

Towards an enumerative geometry of the moduli space of twisted curves and rth roots

AbstractThe enumerative geometry of rth roots of line bundles is crucial in the theory of r-spin curves and occurs in the calculation of Gromov–Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In a previous paper, we constructed a compact moduli stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford’s formula for the Chern character of the Hodge bundle to the direct image of the universal rth root in K-theory.