Linear Systems of Plane Curves with a Composite Number of Base Points of Equal Multiplicity

Linear Systems of Plane Curves with Base Points of Equal Multiplicity

In this article we address the problem of computing the dimension of the space of plane curves of degree $d$ with $n$ general points of multiplicity $m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple $(-1)$-curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed in a previous artic…

Analytic classification of plane branches up to multiplicity 4

We perform the analytic classification of plane branches of multiplicity less or equal than four.

Birationally rigid complete intersections with a singular point of high multiplicity

We prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of these varieties are equal, and they are non-rational. The proof is based on the techniques of the method of maximal singularities, the generalized $4n^2$-inequality for complete intersection singularities and the technique of hypertangent divisors.

超椭圆曲线超特殊性的同时方程的多重性一定理

The multiplicity-one theorem for the superspeciality of hyperelliptic curves

本文通过Lauricella系统类型D推广了超特殊性同时方程的多重性一定理，适用于任意亏格的超椭圆曲线。

Euler number of the compactified Jacobian and multiplicity of rational curves

We show that the Euler number of the compactified Jacobian of a rational curve $C$ with locally planar singularities is equal to the multiplicity of the $δ$-constant stratum in the base of a semi-universal deformation of $C$. In particular, the multiplicity assigned by Yau, Zaslow and Beauville to a rational curve on a K3 surface $S$ coincides with the multiplicity of the normalisation map in the moduli space of stable maps to $S$.

Multiplicity-Free Gonality on Graphs

The divisorial gonality of a graph is the minimum degree of a positive rank divisor on that graph. We introduce the multiplicity-free gonality of a graph, which restricts our consideration to divisors that place at most \(1\) chip on each vertex. We give a sufficient condition in terms of vertex-connectivity for these two versions of gonality to be equal; and we show that no function of gonality can bound multiplicity-free gonality, even for simple graphs. We also prove that multiplicity-free gonality is NP-hard to compute, while still determining it for graph families for which gonality is cu…