Counting Bitangents with Stable Maps

关于多纤维簇的计数

On counting plurifibered varieties

该研究聚焦于多纤维簇映射的计数这一数学代数几何领域的问题，对相关枚举问题展开探讨。

Maps, sheaves, and K3 surfaces

The conjectural equivalence of curve counting on Calabi-Yau 3-folds via stable maps and stable pairs is discussed. By considering Calabi-Yau 3-folds with K3 fibrations, the correspondence naturally connects curve and sheaf counting on K3 surfaces. New results and conjectures (with D. Maulik) about descendent integration on K3 surfaces are announced. The recent proof of the Yau-Zaslow conjecture is surveyed. The paper accompanies my lecture at the Clay research conference in Cambridge, MA in May 2008.

On polynomials counting essentially irreducible maps

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice point in the moduli space of genus-$g$ curves with $n$ labeled points and is given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ in the face degrees $2\ell_1, \ldots, 2\ell_n$. We generalize this by restricting to genus-$g$ maps that are essentially $2b$-irreducible for $b\geq 0$, which loosely speaking means that they are not allowed to possess …

Curve counting in genus one: elliptic singularities & relative geometry

We construct and study the reduced, relative, genus one Gromov--Witten theory of very ample pairs. These invariants form the principal component contribution to relative Gromov--Witten theory in genus one and are relative versions of Zinger's reduced Gromov--Witten invariants. We relate the relative and absolute theories by degeneration of the tangency conditions, and the resulting formulas generalise a well-known recursive calculation scheme put forward by Gathmann in genus zero. The geometric input is a desingularisation of the principal component of the moduli space of genus one logarithmic…

Counting isolated points outside the image of a polynomial map

We consider a generic family of polynomial maps $f:=(f_1,f_2):\mathbb{C}^2\rightarrow\mathbb{C}^2$ with given supports of polynomials, and degree $ d(f):=\max (deg f_1, deg f_2)$. We show that the (non-) properness of maps $f$ in this family depends uniquely on the pair of supports and that the set of isolated points in $\mathbb{C}^2\setminus f(\mathbb{C}^2)$ has a size of at most $6 d(f)$. This improves an existing upper bound $(d(f) - 1)^2$ proven by Jelonek. Moreover, for each $n\in\mathbb{N}$, we construct a dominant map $f$ above, with $d(f) = 2n+2$, and having $2n$ isolated points in $\m…

Counting Higher Genus Curves with Crosscaps in Calabi-Yau Orientifolds

We compute all loop topological string amplitudes on orientifolds of local Calabi-Yau manifolds, by using geometric transitions involving SO/Sp Chern-Simons theory, localization on the moduli space of holomorphic maps with involution, and the topological vertex. In particular we count Klein bottles and projective planes with any number of handles in some Calabi-Yau orientifolds.

Counting rational curves of arbitrary shape in projective spaces

We present an approach to a large class of enumerative problems concerning rational curves in projective spaces. This approach uses analysis to obtain topological information about moduli spaces of stable maps. We demonstrate it by enumerating one-component rational curves with a triple point or a tacnodal point in the three-dimensional projective space and with a cusp in any projective space.

13/2 ways of counting curves

In the past 20 years, compactifications of the families of curves in algebraic varieties X have been studied via stable maps, Hilbert schemes, stable pairs, unramified maps, and stable quotients. Each path leads to a different enumeration of curves. A common thread is the use of a 2-term deformation/obstruction theory to define a virtual fundamental class. The richest geometry occurs when X is a nonsingular projective variety of dimension 3. We survey here the 13/2 principal ways to count curves with special attention to the 3-fold case. The different theories are linked by a web of conjectura…

Counting Plane Rational Curves: Old and New Approaches

These notes are intended as an easy-to-read supplement to part of the background material presented in my talks on enumerative geometry. In particular, the numbers $n_3$ and $n_4$ of plane rational cubics through eight points and of plane rational quartics through eleven points are determined via the classical approach of counting curves. The computation of the latter number also illustrates my topological approach to counting the zeros of a fixed vector bundle section that lie in the main stratum of a compact space. The arguments used in the computation of the number $n_4$ extend easily to co…

Counting rational curves with multiple points and Gromov-Witten invariants of blow-ups

We study Gromov-Witten invariants on the blow-up of P^n at a point, which is probably the simplest example of a variety whose moduli spaces of stable maps do not have the expected dimension. It is shown that many of these invariants can be interpreted geometrically on P^n as certain numbers of rational curves having a multiple point of given order at the blown up point. Moreover, all these invariants can actually be calculated, giving enumerative invariants of P^n which have not been known before.

分圆栈上的高度模空间与函数域上椭圆曲线的计数

Height moduli on cyclotomic stacks and counting elliptic curves over function fields

本文利用twisted stable maps和stacky height functions理论，构造了参数化cyclotomic stacks上固定高度有理点的有限型模空间，解释了函数域上椭圆曲线计数中的低阶主项，回答了Venkatesh的问题。

Counting real curves with passage/tangency conditions

We study the following question: given a set P of 3d-2 points and an immersed curve G in the real plane R^2, all in general position, how many real rational plane curves of degree d pass through these points and are tangent to this curve. We count each such curve with a certain sign, and present an explicit formula for their algebraic number. This number is preserved under small regular homotopies of a pair (P, G), but jumps (in a well-controlled way) when in the process of homotopy we pass a certain singular discriminant. We discuss the relation of such enumerative problems with finite type i…

通过热带几何的二次加丰富平面曲线计数

Quadratically Enriched Plane Curve Counting via Tropical Geometry

本文通过热带几何中的地板图技术，证明光滑环面del Pezzo曲面中经过有理点和共轭点的有理曲线的二次加丰富计数可由特定多重性的垂直拉伸点条件的热带稳定映射计数确定，并给出计算算法。

The Moduli Space of Polynomial Maps and Their Holomorphic Indices: I. Generic Properties in the Case of Having Multiple Fixed Points

Following the author's previous works, we continue to consider the problem of counting the number of affine conjugacy classes of polynomials of one complex variable when its unordered collection of holomorphic fixed point indices is given. The problem was already solved completely in the case that the polynomials have no multiple fixed points, in the author's previous papers. In this paper, we consider the case of having multiple fixed points, and obtain the formulae for generic unordered collections of holomorphic fixed point indices, for each given degree and for each given number of fixed p…