Related papers: Enumerativity of virtual Tevelev degrees

Let $G$ be a finite abelian group and let $K$ be an algebraically closed field of characteristic 0. We consider associative unital algebras $A$ over $K$ graded by $G$, that is $A=\oplus_{g\in G} A_g$, where the vector subspaces $A_g$…

Let $M$ be the moduli space of rank $2$ stable bundles with fixed determinant of degree $1$ on a smooth projective curve $C$ of genus $g\ge 2$. When $C$ is generic, we show that any elliptic curve on $M$ has degree (respect to…

Let $\xi$ be an analytic bracket-generating distribution. We show that the subspace of germs that are singular (in the sense of Control Theory) has infinite codimension within the space of germs of smooth curves tangent to $\xi$. We…

Kontsevich and Manin gave a formula for the number $N_e$ of rational plane curves of degree $e$ through $3e-1$ points in general position in the plane. When these $3e-1$ points have coordinates in the rational numbers, the corresponding set…

The problem of map enumeration concerns counting connected spatial graphs, with a specified number $j$ of vertices, that can be embedded in a compact surface of genus $g$ in such a way that its complement yields a cellular decomposition of…

We show that a holomorphic two-form $\theta$ on a smooth algebraic variety X localizes the virtual fundamental class of the moduli of stable maps $\mgn(X,\beta)$ to the locus where $\theta$ degenerates; it then enables us to define the…

We determine the Gromov-Witten invariants of the local Enriques surfaces for all genera and curve classes and prove the Klemm-Mari\~{n}o formula. In particular, we show that the generating series of genus $1$ invariants of the Enriques…

The Gieseker-Uhlenbeck morphism maps the Gieseker moduli space of stable rank-2 sheaves on a smooth projective surface to the Uhlenbeck compactification, and is a generalization of the Hilbert-Chow morphism for Hilbert schemes of points.…

We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolov's inequality for…

These are notes of lectures given at the NATO Summer School, Montreal 1995. Taubes's recent spectacular work setting up a correspondence between $J$-holomorphic curves in symplectic 4-manifolds and solutions of the Seiberg-Witten equations…

We investigate the Gopakumar-Vafa (GV) theory of local curves, namely, the total spaces of rank two vector bundles with canonical determinant on smooth projective curves. Under a certain genericity condition on the rank two bundles, we…

Let $X$ be an irreducible projective variety of dimension $n$ in a projective space and let $x$ be a point of $X$. Denote by ${\rm Curves}_d(X,x)$ the space of curves of degree $d$ lying on $X$ and passing through $x$. We will show that the…

In this short mostly expository note, we sketch a program for gauging fully extended topological field theories in 3 dimensions. One begins with the spherical fusion category with which one wants to do Levin-Wen or Turaev-Viro. One then…

We study the enumerative geometry of algebraic curves on abelian surfaces and threefolds. In the abelian surface case, the theory is parallel to the well-developed study of the reduced Gromov-Witten theory of K3 surfaces. We prove complete…

In this paper, we propose another characterization of the generalized mirror transformation on the quantum cohomology rings of general type projective hypersurfaces. This characterics is useful for explicit determination of the form of the…

We describe genus g>1 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In Gromov-Witten theory, it becomes a conjecture expressing higher genus…

We remark on the claim that the string-net model of Levin and Wen is a microscopic Hamiltonian formulation of the Turaev-Viro topological quantum field theory. Using simple counterexamples we indicate where interesting extra structure may…

Consider a component of the Hilbert scheme whose general point corresponds to a degree d genus g smooth irreducible and nondegenerate curve in a projective variety X. We give lower bounds for the dimension of such a component when X is P^3,…

We express the genus-two fixed-complex-structure enumerative invariants of P^2 and P^3 in terms of the genus-zero enumerative invariants. The approach is to relate each genus-two fixed-complex-structure enumerative invariant to the…

The ring of invariant polynomials ${\mathbb C}[V]^G$ over a given finite dimensional representation space $V$ of a complex reductive group $G$ is known, by a famous theorem of Hilbert, to be finitely generated. The general proof being…