Related papers: Generalized Tevelev degrees of $\mathbb{P}^1$

Consider the 1-dimensional Hurwitz space parameterizing covers of P^1 branched at four points. We study its intersection with divisor classes on the moduli space of curves. As an application, we calculate the slope of the Teichmuller curve…

In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface $\fn$. We compute the number of such curves through the appropriate number of fixed general points on $\fn$, and the number of…

We use twisted stable maps to answer the following question. Let E\subset P^2 be a smooth cubic. How many rational degree d curves pass through a general points of E, have b specified tangencies with E and c unspecified tangencies, and pass…

We obtain an explicit expression for the number of ramified coverings of the sphere by the torus with given ramification type for a small number of ramification points, and conjecture this to be true for an arbitrary number of ramification…

We construct invariants for any closed semipositive symplectic manifold which count rational curves satisfying tangency constraints to a local divisor. More generally, we introduce invariants involving multibranched local tangency…

We establish an arithmetic intersection theory in the framework of Arakelov geometry over adelic curves. To each projective scheme over an adelic curve, we associate a multi-homogenous form on the group of adelic Cartier divisors, which can…

In this paper we obtain an explicit formula for the number of curves in a compact complex surface $X$ (passing through the right number of generic points), that has up to one node and one singularity of codimension $k$, provided the total…

Given a Galois cover of curves X to Y with Galois group G which is totally ramified at a point x and unramified elsewhere, restriction to the punctured formal neighborhood of x induces a Galois extension of Laurent series rings…

We study Shimura curves of PEL type in $\mathsf{A}_g$ generically contained in the Prym locus. We study both the unramified Prym locus, obtained using \'etale double covers, and the ramified Prym locus, corresponding to double covers…

Generalized geometry finds many applications in the mathematical description of some aspects of string theory. In a nutshell, it explores various structures on a generalized tangent bundle associated to a given manifold. In particular,…

Several generalizations of Vershik-Kerov limit shape problem are motivated by topological string theory and supersymmetric gauge theory instanton count. In this paper specifically we study the circular and linear quiver theories. We also…

In this paper we further develop the theory of geometric tropicalization due to Hacking, Keel and Tevelev and we describe tropical methods for implicitization of surfaces. More precisely, we enrich this theory with a combinatorial formula…

Given a finite field $\mathbb{F}_{q}$, we study the distribution of the number of $\mathbb{F}_{q}$-points on (possibly singular) affine curves given by the polynomial equations of the form $C_{f} : y^{m} = f(x)$, where $f$ is randomly…

We generalize Siegel's theorem on integral points on affine curves to integral points of bounded degree, giving a complete characterization of affine curves with infinitely many integral points of degree d or less over some number field.…

We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{CP}^2$ that pass through $3d+1-m$ generic points and that have an $m$-fold singular point. The special case of counting curves with a triple point was…

We show that the tangent bundle of a projective manifold with nef anticanonical class is generically nef. That is, its restriction to a curve cut out by general sufficiently ample divisors is a nef vector bundle. This confirms a conjecture…

Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$. We show that if $|R| < \frac{3}{2}n$ and $P…

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…

In this short note, we investigate the generalization of Lehmer's problem to finitely generated fields over $\mathbb{Q}$.

For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various…