Related papers: Level-$\delta$ limit linear series

A vertex coloring of a simplicial complex $\Delta$ is called a linear coloring if it satisfies the property that for every pair of facets $(F_1, F_2)$ of $\Delta$, there exists no pair of vertices $(v_1, v_2)$ with the same color such that…

Over the past 20 years, a great deal of work has been done on the moduli spaces of coherent systems on algebraic curves. Until recently, however, there has been very little work on the fixed determinant case, except for the special case of…

In this article we construct a categorical resolution of singularities of an excellent reduced curve $X$, introducing a certain sheaf of orders on $X$. This categorical resolution is shown to be a recollement of the derived category of…

A line bundle L on a smooth curve X is nonspecial if and only if L admits a presentation L=K_X -D +E for some effective divisors D and E>0 on X with gcd (D, E)=0 and h^0 (X, O_X (D))=1. In this work, we define a minimal presentation of L…

We define a geometrically meaningful compactification of the moduli space of smooth plane curves, which can be calculated explicitly. The basic idea is to regard a plane curve D in P^2 as a pair (P^2,D) of a surface together with a divisor,…

We derive bounds on the path length $\zeta$ of gradient descent (GD) and gradient flow (GF) curves for various classes of smooth convex and nonconvex functions. Among other results, we prove that: (a) if the iterates are linearly convergent…

Let $\Delta$ be a connected, pure $2$-dimensional simplicial complex embedded in $\mathbb{R}^2$ and let $C^{r}(\hat{\Delta})$ be the homogenized spline module of $\Delta$ with smoothness $r$. To study $C^{r}(\hat{\Delta})$, Schenck and…

A convex lattice polygon Delta determines a pair (S,L) of a toric surface together with an ample toric line bundle on S. The Severi degree N^{Delta,delta} is the number of delta-nodal curves in the complete linear system |L| passing through…

In this paper we mainly describe $\mathbb{Q}$-Gorenstein smoothings of projective surfaces with only Wahl singularities which have birational fibers. For instance, these degenerations appear in normal degenerations of the projective plane,…

Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. The degree graph $\Delta(G)$ of $G$ is defined as the simple undirected graph whose vertex set ${\rm{V}}(G)$ consists…

In most cases the semigroup at infinity $S$ of a curve $C$ with only one place at infinity is generated by a $\delta$-sequence. This sequence provides geometrical information on $C$ such as the dual graph of the resolution of the…

A theorem of Wiegerinck asserts that the Bergman space of an open subset of the complex numbers is either infinite-dimensional or trivial. Recently, this has been generalized to holomorphic vector bundles over the projective line by the…

We study ample divisors X with only rational singularities on abelian varieties that decompose into a sum of two lower dimensional subvarieties, X=V+W. For instance, we prove an optimal lower bound on the degree of the corresponding…

It is known that to every 1|1 dimensional supercurve X there is associated a dual supercurve \hat{X}, and a superdiagonal \Delta in their product. We establish that the categories of D-modules on X, \hat{X}, and \Delta are equivalent. This…

Associated to an open subgroup $G$ of $\GL_2(\Zhat)$ satisfying conditions $-I \in G$ and $\det(G) \subsetneq (\Zhat)^{\times}$ there is a modular curve $X_G$ which is a smooth compact curve defined over an extension of $\Q.$ In this…

Given a smooth del Pezzo surface $X_d \subseteq \mathbb{P}^{d}$ of degree $d,$ we show that a smooth irreducible curve $C$ on $X_d$ represents the first Chern class of an Ulrich bundle on $X_d$ if and only if its kernel bundle $M_C$ admits…

This text is a presentation of a set of formulae, first found by Vainsencher (for $\delta \leq 6$) and shortly after improved by Kleiman and Piene, counting $\delta$-nodal curves in a complete linear system on a smooth surface, if $\delta…

In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar curvature should have a subsequence which converges in some weak sense to a limit space with some generalized notion of nonnegative scalar curvature.…

The past decade has witnessed two important new developments in the study of linear series on algebraic varieties. First, vector bundles have emerged as powerful tools for analyzing linear series on curves and surfaces. More recently, the…

In this paper we prove, using a refinement of Terracini's Lemma, a sharp lower bound for the degree of (higher) secant varieties to a given projective variety, which extends the well known lower bound for the degree of a variety in terms of…