相关论文: Enumerative geometry of hyperelliptic plane curves

This paper studies the global structure of algebraic curves defined by generalized unitarity cut of four-dimensional three-loop diagrams with eleven propagators. The global structure is a topological invariant that is characterized by the…

By showing a structural result on how the Rozansky-Witten invariants of Hilbert schemes of points on a K3 surface and the Generalised Kummer varieties depend on the dimension, we give a method how to compute a large class of Rozansky-Witten…

These expository notes are based on a series of lectures given at the May 2018 Snowbird workshop, Crossing the Walls in Enumerative Geometry. We give an introductory treatment of the notion of a virtual fundamental class in algebraic…

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…

We first present the construction of the moduli space of real pseudo-holomorphic curves in a given real symplectic manifold. Then, following the approach of Gromov and Witten, we construct invariants under deformation of real rational…

This paper contains an attempt to formulate rigorously and to check predictions in enumerative geometry of curves following from Mirror Symmetry. The main tool is a new notion of stable map. We give an outline of a contsruction of…

We give a formula for the number of genus-two fixed-complex-structure degree-d plane curves passing through 3d-2 points in general position. This is achieved by completing Katz-Qin-Ruan's approach. This paper's formula agrees with the one…

Enumerative geometry, the art and science of counting geometric objects satisfying geometric conditions, has seen a resurgence of activity in recent years due to an influx of new techniques that allow for enriched computations. This paper…

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r$. In this…

In this paper we obtain an explicit formula for the number of curves in two dimensional complex projective space, of degree d, passing through d(d+3)/2-(k+1) generic points and having one node and one codimension k singularity, where k is…

The geometric Tevelev degrees of projective space enumerate general, pointed algebraic curves interpolating through the maximal possible number of points. Previous work expresses these invariants in terms of Schubert calculus. Extending…

Recently, there has been great interest in the application of composition laws to problems in enumerative geometry. Using the moduli space of stable maps, we compute the number of irreducible, reduced, nodal, degree-$d$ genus-$2$ plane…

Using reduced Gromov-Witten theory, we define new invariants which capture the enumerative geometry of curves on holomorphic symplectic 4-folds. The invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds,…

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable twisted rational curves of degree $d$ in ${\Bbb P}^3\backslash {\Bbb P}^1$ is isomorphic to a…

Bielliptic surfaces appear as quotient of a product of two elliptic curves and were classified by Bagnera-Franchis. We give a concrete way of computing their GW-invariants with point insertions using a floor diagram algorithm. Using the…

We investigate the geometry of smooth hyperelliptic curves that possess additional involutions, especially from the point of view of the Prym theory. Our main result is the injectivity of the Prym map for hyperelliptic…

We consider the moduli space $\Hh_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$. In order to get cohomological information we wish to make $\s_n$-equivariant counts of the numbers of points defined over finite fields of…

This paper is an elementary introduction to the theory of moduli spaces of curves and maps. As an application to enumerative geometry, we show how to count the number of bitangent lines to a projective plane curve of degree $d$ by doing…

This paper describes the structure of the moduli space of holomorphic curves and constructs Gromov Witten invariants in the category of exploded manifolds. This includes defining Gromov Witten invariants relative to normal crossing divisors…

This paper introduces a new way of generalizing Hilbert's two-dimensional space-filling curve to arbitrary dimensions. The new curves, called harmonious Hilbert curves, have the unique property that for any d' < d, the d-dimensional curve…