Justyna Szpond
In this note we compute values of global linear Harbourne constants over arbitrary fields for up to ten lines. These invariants have appeared recently in the discussions around the Bounded Negativity Conjecture. They seem to be of independent interest also from the point of view of combinatorics.
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In the present note we study absolute linear Harbourne constants. These are invariants which were introduced in order to relate the lower bounds on the selfintersection of negative curves on birationally equivalent surfaces to the complexity of the birational map between them. We provide various lower and upper bounds on Harbourne constants and give their values for the number of lines $s$ of the form $p^{2r}+p^r+1$ for any prime number $p$ and also for all values of $s$ up to $31$. This extends considerably earlier results of the third author.
In this note we exhibit the so-called Harbourne constants which capture and measure the Bounded Negativity on various birational models of an algebraic surface. We show an estimation for Harbourne constants for conic configurations on the complex projective plane with transversal intersection points.
In this note we find a bound for the so-called global linear Harbourne constants for smooth hypersurfaces in $\mathbb{P}^{3}_{\mathbb{C}}$
Let $X$ be a smooth projective surface and let $\mathcal{C}$ be an arrangement of curves on $X$. The Harbourne constant of $\mathcal{C}$ was defined as a way to investigate the occurrence of curves of negative self-intersection on blow ups of $X$. This is related to the bounded negativity conjecture which predicts that the self-intersection number of all reduced curves on a surface is bounded below by a constant. We consider a geometrically ruled surface $X$ over a smooth curve and give lower bounds for the Harbourne constants of transversal arrangements of curves on $X$.
We construct and study curves with low H-constants on abelian and K3 surfaces. Using the Kummer $(16_{6})$-configurations on Jacobian surfaces and some $(16_{10})$-configurations of curves on $(1,3)$-polarized Abelian surfaces, we obtain examples of configurations of curves of genus $>1$ on a generic Jacobian K3 surface with H-constants $<-4$.
We describe the effect of ramified morphisms on Harbourne constants of reduced effective divisors. With this goal, we introduce the pullback of a weighted cluster of infinitely near points under a dominant morphism between surfaces, and describe some of its basic properties. As an application, we describe configurations of curves with transversal intersections and $H$-index arbitarily close to $-25/7\simeq -3.571$, smaller than any previously known result.
In this note we relate the SHGH Conjecture to the rationality of one-point Seshadri constants on blow ups of the projective plane, and explain how rationality of Seshadri constants can be tested with the help of functions on Newton--Okounkov bodies.
Bounded negativity, Harbourne constants and transversal arrangements of curves
本文将Harbourne常数的研究从直线、圆锥曲线和三次曲线构型推广到任意次数曲线构型的爆破，并研究了非负Kodaira维数曲面上充分正曲线的横截构型。
Let $Σ$ be a finite collection of linear forms in $\mathbb K[x_0,\ldots,x_n]$, where $\mathbb K$ is a field. Denote ${\rm Supp}(Σ)$ to be the set of all nonproportional elements of $Σ$, and suppose ${\rm Supp}(Σ)$ is generic, meaning that any $n+1$ of its elements are linearly independent. Let $1\leq a\leq |Σ|$. In this article we prove the conjecture that the ideal generated by (all) $a$-fold products of linear forms of $Σ$ has linear graded free resolution. As a consequence we prove the Geramita-Harbourne-Migliore conjecture concerning the primary decomposition of ordinary powers of defining…
Similarly to the linear Harbourne constant recently defined, we study the elliptic $H$-constants of $\mathbb{P}^{2}$ and Abelian surfaces. We exhibit configurations of smooth plane cubic curves whose Harbourne index is arbitrarily close to $-4$. As a Corollary, we obtain that the global $H$-constant of any surface $X$ is less or equal to $-4$. Related to these problems, we moreover give a new inequality for the number and multiplicities of singularities of elliptic curves arrangements on Abelian surfaces, inequality which has a close similarity to the one of Hirzebruch for arrangements of line…
Boundedness Results for Planar Linear Systems Assuming The Segre-Harbourne-Gimigliano-Hirschowitz Conjecture
本文在Segre-Harbourne-Gimigliano-Hirschowitz猜想成立的前提下，研究胀开$n\geq10$个一般点的射影平面上的完全线性系统，完成相关分类并证明有界性等结果。
In the paper we compute the virtual dimension (defined by the Hilbert polynomial) of a space of hypersurfaces of given degree containing $s$ codimension 2 general linear subspaces in $\mathbb{P}^n$. We use Veneroni maps to find a family of unexpected hypersurfaces (in the style of B. Harbourne, J. Migliore, U. Nagel, Z. Teitler) and rigorously prove and extend examples presented in the paper by B. Harbourne, J. Migliore and H. Tutaj-Gasińska.
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.
Quasi-Homogeneous Linear Systems on P^2 with Base Points of Multiplicity 6
本文证明了P^2上重数为6的拟齐次线性系的Harbourne-Hirschowitz猜想。对于证明，我们使用Ciliberto和Miranda的平面退化和Lapace，Seibert，Ugaglia和Yang的结果。作为一个应用程序，我们得到的特殊系统的多重性6的分类。
Quasi-homogeneous linear systems on P2 with base points of multiplicity 7, 8, 9, 10
本文证明了关于$\mathbb P^2$上拟齐次线性系的Harbourne-Hirschowitz猜想，其中$m=7$，8，9，10，即通过一般位置上点的给定次数曲线系，其重数至少为$m,...,m,m_0$，其中$m=7$，8，9，10，$m_0$是任意的.
Linear systems in $\mathbb{P}^2$ with base points of bounded multiplicity
对于基点的重数小于等于7的线性系，利用著名的射影平面的退化和将点专门化到直线上的组合技巧，给出了Harbourne-Hirschowitz猜想的一个证明.
Characterization of linear spaces of matrices of constant rank from syzygy bundles
本文基于syzygy丛刻画了常数秩线性形式矩阵，证明其与部分线性分解相关联，并列出了秩不超过7的不可分解矩阵及简单向量丛的模空间。
On linear spaces of skew-symmetric matrices of constant rank
我们描述了六阶常秩四的复反对称矩阵的射影平面空间。我们证明了它有四个连通分支，在PGL_6的作用下都是26维的齐次连通分支。
Matrices of linear forms of constant rank from vector bundles on projective spaces
本文利用射影空间上小第一陈类的全局生成向量丛分类，完全刻画了秩不超过六的不可分解常秩线性形式矩阵，并揭示了其与Steiner丛、Drézet丛、刚性齐次向量丛及instanton丛的联系。
We present a new method to study 4-dimensional linear spaces of skew-symmetric matrices of constant co-rank 2, based on rank 2 vector bundles on P^3 and derived category tools. The method allows one to prove the existence of new examples of size 10x10 and 14x14 via instanton bundles of charge 2 and 4 respectively, and provides an explanation for what used to be the only known example (Westwick 1996). We also give an algorithm to construct explicitly a matrix of size 14 of this type.
Mixed motives and linear forms in the Catalan constant
本文通过构造混合动机，将Catalan常数作为周期，并利用该动机获得1和G的线性形式及其系数。
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