Linear Systems of Plane Curves with Base Points of Equal Multiplicity

Special homogeneous linear systems on Hirzebruch surfaces

The Segre-Gimigliano-Harbourne-Hirschowitz Conjecture can be naturally formulated for Hirzebruch surfaces F_n. We show that this Conjecture holds for imposed base points of equal multiplicity bounded by 8.

舒伯特除子的无重数乘积

Multiplicity-free products of Schubert divisors

研究任意域上旗簇中舒伯特除子类无重数乘积在 Chow 环中的次数，能将其分解为舒伯特簇类线性组合，通过研究组合系数给出判别准则，确定系数情况。

Rank two vector bundles on polarised Halphen surfaces and the Gauss-Wahl map for du Val curves

A genus-g du Val curve is a degree-3g plane curve having 8 points of multiplicity g, one point of multiplicity g-1, and no other singularity. We prove that the corank of the Gauss-Wahl map of a general du Val curve of odd genus (>11) is equal to one. This, together with the results of [1], shows that the characterisation of Brill-Noether-Petri curves with non-surjective Gauss-Wahl map as hyperplane sections of K3 surfaces and limits thereof, obtained in [3], is optimal.

Bi-Lipschitz invariance of the multiplicity

The multiplicity of an algebraic curve $C$ in the complex plane at a point $p$ on that curve is defined as the number of points that occur at the intersection of $C$ with a general complex line that passes close to the point $p$. It is shown that $p$ is a singular point of the curve $C$ if and only if this multiplicity is greater than or equal to 2, in this sense, such an integer number can be considered as a measure of how singular can be a point of the curve $C$. In these notes, we address the classical concept of multiplicity of singular points of complex algebraic sets (not necessarily com…

Reduction method for linear systems of plane curves with base fat points

In the paper we develop a new method of proving non-speciality of a linear system with base fat points in general position. Using this method we show that the Hirschowitz-Harbourne Conjecture holds for systems with base points of equal multiplicity bounded by 42.

Quasi-homogeneous linear systems on $\mathbb{P}^2$ with base points of multiplicity 5

In this paper we consider linear systems of $\mathbb{P}^2$ with all but one of the base points of multiplicity 5. We give an explicit way to evaluate the dimensions of such systems.

关于平面曲线的Saito基

On the Saito basis for plane curves

本文研究解析平面曲线的Saito模与挠子模，提出计算方法，并用于计算重数不超过三的曲线的解析不变量。

Quasi-Homogeneous Linear Systems on P^2 with Base Points of Multiplicity 6

In this paper we prove the Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems of multiplicity 6 on P^2. For the proof we use the degeneration of the plane by Ciliberto and Miranda and results by Laface, Seibert, Ugaglia and Yang. As an application we derive a classification of the special systems of multiplicity 6.

关于具有最大数量最大切线的四次曲线

On quartics with the maximal number of the maximal tangency lines

本文研究了Fermat和Komiya-Kuribayashi四次曲线上最大切线的直线排列与点构型，并进一步分析了其六切点及相关圆锥曲线构型。

Diminished Fermat-type arrangements and unexpected curves

The purpose of this note is to present and study a new series of the so-called unexpected curves. They enjoy a surprising property to the effect that their degree grows to infinity, whereas the multiplicity at a general fat point remains constant, equal $3$, which is the least possible number appearing as the multiplicity of an unexpected curve at its singular point. We show that additionally the BMSS dual curves inherits the same pattern of behaviour.

On Serre Intersection Multiplicity Conjecture

In this short note, we expose some of the works on Serre intersection multiplicity conjecture. I provide a proof of the vanishing of Serre intersection multiplicity in non-proper intersection over a regular ring based on the intersection theory in W. Fulton.

Linear systems over P1xP1 with base points of multiplicity bounded by three

We propose a combinatorial method of proving non-specialty of a linear system of curves with multiple points in general positions. As an application we obtain a classification of special linear systems on P1xP1 for which the multiplicities do not exceed 3.

Cubic threefolds and abelian varieties of dimension five. II

This paper extends joint work with R. Friedman to show that the closure of the locus of intermediate Jacobians of smooth cubic threefolds, in the moduli space of principally polarized abelian varieties (ppav's) of dimension five, is an irreducible component of the locus of ppav's whose theta divisor has a point of multiplicity three or more. This paper also gives a sharp bound on the multiplicity of a point on the theta divisor of an irreducible ppav of dimension less than or equal to five; for dimensions four and five, this improves the bound due to J. Kollár, R. Smith-R. Varley, and L. Ein-R…

胖点的弧空间的重数结构

Multiplicity structure of the arc space of a fat point

本文研究直线上由$x^m=0$定义的胖点的弧空间，得到其截断弧代数维数的生成级数与定义理想的字典序首项理想，证明了Afsharijoo的相关猜想，关联了微分代数重数等多个方向。

Linear systems in $\mathbb{P}^2$ with base points of bounded multiplicity

We present a proof of the Harbourne-Hirschowitz conjecture for linear systems with base points of multiplicity seven or less. This proof uses a well-known degeneration of the projective plane, as well as a combinatorial technique that arises from specializing points onto a line.

Positivity of Intersection Multiplicity Over a Two-Dimensional Base

We show that Serre's Intersection Multiplicity Conjecture holds for a formal power series ring A over a complete, two-dimensional regular local ring R. From this, we deduce the corresponding result for the local rings of any scheme X which is a smooth extension of a regular, two-dimensional base Y. We also investigate the connection between intersection multiplicity and transversality in the unramified setting via a local analysis on the blowup.