Related papers: On the Hartshorne-Hirschowitz theorem

We formulate a tropical analogue of Grothendieck's section conjecture: that for every stable graph G of genus g>2, and every field k, the generic curve with reduction type G over k satisfies the section conjecture. We prove many cases of…

We consider a semistable degeneration of K3 surfaces, equipped with an effective divisor that defines a polarisation of degree two on a general fibre. We show that the map to the relative log canonical model of the degeneration maps every…

In this paper we prove the equivalence of two conjectures on linear systems through fat points on a generic K3 surface. The first conjecture is exactly as Segre conjecture on the projective plane. Whereas the second characterizes such…

We construct a collection of higher Chow cycles on certain surfaces which degenerate to an arrangement of planes in general position. When its degree is 4, this construction gives a new explicit proof of the Hodge-D-Conjecture for a certain…

For the universal isomonodromic deformation of an irreducible logarithmic rank two connection over a smooth complex projective curve of genus at least two, consider the family of holomorphic vector bundles over curves underlying this…

One version of the classical Lefschetz hyperplane theorem states that for $U \subset \mathbb P^n$ a smooth quasi-projective variety of dimension at least $2$, and $H \cap U$ a general hyperplane section, the resulting map on \'etale…

Given a planar straight-line graph $G=(V,E)$ in $\mathbb{R}^2$, a \emph{circumscribing polygon} of $G$ is a simple polygon $P$ whose vertex set is $V$, and every edge in $E$ is either an edge or an internal diagonal of $P$. A circumscribing…

Inspired by Bondal's conjecture, we study the behavior of exceptional sequences of line bundles on rational C*-surfaces under homogeneous degenerations. In particular, we provide a sufficient criterion for such a sequence to remain…

We show that the number of lines in an $m$--homogeneous supersolvable line arrangement is upper bounded by $3m-3$ and we classify the $m$--homogeneous supersolvable line arrangements with two modular points up-to lattice-isotopy. A lower…

We show that the space of orthogonally separable coordinates on the sphere $S^3$ induces a natural family of integrable systems, which after symplectic reduction leads to a family of integrable systems on $S^2 \times S^2$. The generic…

Let $X$ be a hypersurface of degree $d$ in $\Bbb P^n$ and $F_X$ be the scheme of $\Bbb P^r$'s contained in $X$. If $X$ is generic, then $F_X$ will have the expected dimension (or empty) and its class in the Chow ring of $G(r+1,n+1)$ is…

We prove that the 1D Schr\"odinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>2/3$ for small $L^{2}$ data. The result follows from an application of the ``I-method''. This method allows to…

We propose a strengthening of the Grothendieck--Lefschetz hyperplane theorem for the local Picard group, prove some special cases and derive several consequences to the deformation theory of log canonical singularities. Version 2: Main…

We introduce a notion of homogeneous topological order, which is obeyed by most, if not all, known examples of topological order including fracton phases on quantum spins (qudits). The notion is a condition on the ground state subspace,…

In Euclidean $3$-space, it is well known that the Sine-Gordon equation was considered in the nineteenth century in the course of investigations of surfaces of constant Gaussian curvature $K=-1$. Such a surface can be constructed from a…

We prove that, for 3 < m < n-1, the Grassmannian of m-dimensional subspaces of the space of skew-symmetric forms over a vector space of dimension n is birational to the Hilbert scheme of the degeneracy loci of m global sections of Omega(2),…

It is well known since Noether that the gonality of a smooth plane curve of degree d>3 is d-1. Given a k-dimensional complex projective variety X, the most natural extension of gonality is probably the degree of irrationality, that is the…

As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of…

A full theory for hinged beams and degenerate plates with multiple intermediate piers is developed. The analysis starts with the variational setting and the study of the linear stationary problem in one dimension. Well-posedness results are…

We explicitly describe the KSBA/Hacking compactification of a moduli space of log surfaces of Picard rank 2. The space parametrizes log pairs $(S, D)$ where $S$ is a degeneration of $\mathbb{P}^1 \times \mathbb{P}^1$ and $D \subset S$ is a…