Related papers: Moduli of curves on Enriques surfaces

We define and count lattice points in the moduli space of stable genus g curves with n labeled points. This extends a construction of the second author for the uncompactified moduli space. The enumeration produces polynomials with top…

Consider the family S of irreducible plane curves of degree n with d nodes and k cusps as singularities. Let W be an irreducible component of S. We consider the natural rational map from W to the moduli space of curves of genus…

This note is devoted to the definition of moduli spaces of rational tropical curves with n marked points. We show that this space has a structure of a smooth tropical variety of dimension n-3. We define the Deligne-Mumford compactification…

Let $X$ be a smooth projective Fano variety over the complex numbers. We study the moduli spaces of rational curves on $X$ using the perspective of Manin's Conjecture. In particular, we bound the dimension and number of components of spaces…

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…

We prove that the moduli space of 2-convex embedded n-spheres in R^{n+1} is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques' influential proof of the…

We construct the generic component of the moduli space of the germs of Legendrian curves with generic plane projection topologicaly equivalent to a curve y^n = x^m, (n,m)=1.

We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus $g$, our moduli space is a stacky fan whose cones are indexed…

Let $M$ be a Milnor sphere or, more generally, the total space of a linear $S^3$-bundle over $S^4$ with $H^4(M;\mathbb{Q})=0$. We show that the moduli space of metrics of nonnegative sectional curvature on $M$ has infinitely many path…

We prove that the coarse moduli space of curves of genus 6 is birational to an arithmetic quotient of a bounded symmetric domain of type IV by giving a period map to the moduli space of some lattice-polarized K3 surfaces.

We construct the moduli space of r-jets at a point of Riemannian metrics on a smooth manifold. The construction is closely related to the problem of classification of jet metrics via differential invariants. The moduli space is proved to be…

This work is motivated by two central questions in the birational geometry of moduli spaces of curves -- Fulton's conjecture and the effective cone of $\bar M_g$. We study the algebro-geometric aspect of Teichmuller curves parameterizing…

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion…

We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing…

We show that for an orientable non-spin manifold with fundamental group $\mathbb{Z}_2$ and universal cover $S^2\times S^3,$ the moduli space of metrics of nonnegative sectional curvature has infinitely many path components. The…

We compute the Euler characteristics of the moduli spaces of abelian vortices on curves with nodal and cuspidal singularities. This generalizes our previous work where only nodes were taken into account. The result we obtain is again…

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of bidegree (3, 3) contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the…

It is well known that knots are countable in ordinary knot theory. Recently, knots {\it with intersections} have raised a certain interest, and have been found to have physical applications. We point out that such knots --equivalence…

We give a complete classification of complex Q-homology projective planes with isolated rational double point singularities and numerically trivial canonical bundle. There are 31 types, and each has one-dimensional moduli. In fact, all…

The main purpose of this paper is to prove the existence of the moduli space parameterizing the embedded curve singularities of $(k^N,0)$ with an admissible Hilbert polynomial and to study its basic properties.