Grzegorz Malara, Piotr Pokora, Halszka Tutaj-Gasińska
In this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study $3$-syzygy arrangements and we present examples that admit unexpected curves.
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Free plane curves with a linear Jacobian syzygy
本文研究了具有线性雅可比辛基的自由平面曲线，通过几何方法对希尔伯特-巴尔奇矩阵进行分类，揭示了其结构特性。
Here we consider the set of bundles $\{V_n\}_{n\geq 1}$ associated to the plane trinomial curves $k[x,y,z]/(h)$. We prove that the Frobenius semistability behaviour of the reduction mod $p$ of $V_n$ is a function of the congruence class of $p$ modulo $2λ_h$ (an integer invariant associated to $h$). As one of the consequences of this, we prove that if $V_n$ is semistable in characteristic 0, then its reduction mod $p$ is strongly semistable, for $p$ in a Zariski dense set of primes. Moreover, for any given finitely many such semistable bundles $V_n$, there is a common Zariski dense set of such …
We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreduciblity of unexpected plane curves of a set of points $Z$ in $\mathbb{P}^2$, using the minimal degree of a Jacobian syzygy of the defining equation for the dual line arrangement $\mathcal A_Z$. Several applications of this new approach are given. In particular, we show that the irreducible unexpected quintics may occur only when the set $Z$ has the cardinality equal to 11 or 12, and describe five cases where this happens.
We investigate degenerations of syzygy bundles on plane curves over $p$-adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fiber consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fiber is a syzygy bundle and which become trivial when restricted to each projective line in the special fiber. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamen…
On the instability of syzygy bundles on toric surfaces
本文研究环面上合冲束的斜率稳定性，证明除两类特殊环面外，总可选取极化使高次幂线丛对应的合冲束失稳。
We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom-Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of …
Let $C:f=0$ be a reduced curve in the complex projective plane. The minimal degree $mdr(f)$ of a Jacobian syzygy for $f$, which is the same as the minimal degree of a derivation killing $f$, is an important invariant of the curve $C$, for instance it can be used to determined whether $C$ is free or nearly free. In this note we study the relations of this invariant $mdr(f)$ with a decomposition of $C$ as a union of two curves $C_1$ and $C_2$, without common irreducible components. When all the singularities that occur are quasihomogeneous, a result by Schenck, Terao and Yoshinaga yields finer i…
In previous works, the authors investigated the relationships between linear stability of a generated linear series $|V|$ on a curve $C$, and slope stabillity of the vector bundle $M_{V,L} := \ker (V \otimes \mathcal{O}_C \to L)$. In particular, the second named author and L. Stoppino conjecture that, for a complete linear system $|L|$, linear (semi)stability is equivalent to slope (semi)stability of $M_V$, and the first and third named authors proved that this conjecture holds for hyperelliptic and for generic curves. In this work we provide a counterexample to this conjecture on any smooth p…
Several papers have been written studying unexpected hypersurfaces. We say a finite set of points Z admits unexpected hypersurfaces if a general union of fat linear subspaces imposes less that the expected number of conditions on the ideal of Z. In this paper, we introduce the concept of a very unexpected hypersurface. This is a stronger condition which takes into account an explanation for some hypersurfaces previously considered unexpected. We then develop a duality theory to relate the study of very unexpected hypersurfaces to the derivations of dual hyperplane arrangements. This allows us …
A closed subscheme of codimension two $T \subset P^2$ is a quasi complete intersection (q.c.i.) of type $(a,b,c)$ if there exists a surjective morphism $\mathcal{O} (-a) \oplus \mathcal{O} (-b) \oplus \mathcal{O} (-c) \to \mathcal{I} _T$. We give bounds on deg$(T)$ in function of $a,b,c$ and $r$, the least degree of a syzygy between the three polynomials defining the q.c.i. As a by-product we recover a theorem of du Plessis-Wall on the global Tjurina number of plane curves and some other related results.
We identify several classes of curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible components. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on $\mathbb{P}^2$, is used to get a sharp lower bound for the initial degree of the Jacobian module $N(f)$, under a semistability condition.
The classification of quasi-elliptic fibrations and unexpected plane cubics in characteristics 2 and 3
本文分类了特征2和3域上拟椭圆曲面的同构类及其到射影平面的爆破收缩序列，并据此完整分类了特征2中的意外平面三次曲线，同时证明了特征3中不存在此类曲线。
On arrangements of smooth plane quartics and their bitangents
本文从多视角重新探讨光滑平面四次曲线及其二重切线的几何性质，研究高度对称四次曲线二重切线排列的弱组合学，为Ciani pencil光滑成员的二重切线排列四重交点数建立下界，并构造3-合冲约化平面曲线新例子。
A characterization of quasi-homogeneous singularities of free and nearly free plane curves
本文通过 Jacobian 理想的第一 syzygy 矩阵，给出了自由与近自由平面曲线拟齐次孤立奇点的新刻画。
First we give a sharp upper bound for the cardinal $m$ of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve $C$. Next we discuss the sharpness of an upper bound, given by A. du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve $C$, in terms of its degree $d$ and of the minimal degree $r\leq d-1$ of a Jacobian syzygy. We give a homological characterization of the curves whose global Tjurina number equals the du Plessis-Wall upper bound, which implies in particular that for such curves the upper bound for $m$ is also …
Our research is motivated by recent work of Cook II, Harbourne, Migliore, and Nagel on configurations of points in the projective plane with properties that are unexpected from the point of view of the postulation theory. In this note, we revisit the basic configuration of nine points appearing in work of Gennaro/Ilardi/Vallès and Harbourne, and we exhibit some additional new properties of this configuration. We then pass to projective three-space and exhibit a surface with unexpected postulation properties there. Such higher dimensional phenomena have not been observed so far.
The purpose of this note is to establish a direct link between the theory of unexpected hypersurfaces and varieties with defective osculating behavior. We identify unexpected plane curves of degree 4 as sections of a rational surface X of degree 7 in P^5 with its osculating spaces of order 2 which in every point of X have dimension lower than expected. We put this result in perspective with earlier examples of surfaces with defective osculating spaces due to Shifrin and Togliatti. Our considerations are rendered by an analysis of Lefschetz Properties of ideals associated with the studied surfa…
Jacobian schemes of conic-line arrangements and eigenschemes
该研究针对射影平面圆锥线配置，给出达到全局Tjurina数上下界的几何刻画，并应用于刻画Jacobian概型为三元张量特征概型的配置及极映射几何。
The Bourbaki Degree of Plane Projective Curves
本研究引入Bourbaki次数作为新数值不变量，聚焦既约形式梯度理想的第一合冲模情形，旨在通过该不变量研究平面曲线奇点，为射影平面曲线分类提供新维度并推导其上下界。
In a recent paper arXiv:1602.02300v2, Cook II, Harbourne, Migliore and Nagel related the splitting type of a line arrangement in the projective plane to the number of conditions imposed by a general fat point of multiplicity $j$ to the linear system of curves of degree $j+1$ passing through the configuration of points dual to the given arrangement. If the number of conditions is less than the expected, we say that the configuration of points admits unexpected curves. In this paper, we characterize supersolvable line arrangements whose dual configuration admits unexpected curves and we provide …
The computation of the dimension of linear systems of plane curves through a bunch of given multiple points is one of the most classic issues in Algebraic Geometry. In general, it is still an open problem to understand when the points fail to impose independent conditions. Despite many partial results, a complete solution is not known, even if the fixed points are in general position. The answer in the case of general points in the projective plane is predicted by the famous Segre-Harbourne-Gimigliano-Hirschowitz conjecture. When we consider fixed points in special position, even more interest…
The notion of an unexpected curve in the plane was introduced in 2018, and was quickly generalized in several directions in a flurry of mathematical activity by many authors. In this expository paper we first describe some of the main results on unexpected hypersurfaces. Then we summarize two offshoots of this theory. First we look at sets of points in $\mathbb P^3$ whose general projection is a planar complete intersection (so-called {\it geproci} sets). Although we now know a lot about these sets, much remains mysterious. Then we describe an interesting measure of unexpectedness called {\it …
Let $\mathcal C :f=0$ be a curve arrangement in the complex projective plane. If $\mathcal C$ contains a curve subarrangement consisting of at least three members in a pencil, then one obtains an explicit syzygy among the partial derivatives of the homogeneous polynomial $f$. In many cases this observation reduces the question about the freeness or the nearly freeness of $\mathcal C$ to an easy computation of Tjurina numbers. Some consequences for Terao's conjecture in the case of line arrangements are also discussed as well as the asphericity of some complements of geometrically constructed f…
A minimal resolution for the Jacobian ideal of a generic curve arrangement
本文利用似然对应理论，为复射影平面中节点曲线排列的雅可比理想的第一、第二合冲模构造了显式极小生成元集，并推广到超曲面排列情形。
We study deformations of plane curve singularities from an analytic point of view and obtain some new concrete results. We show some rather unexpected properties of Puiseux coefficients treated as functions on a suitably defined parameter space. The methods used in paper are very elementary.
Unexpected Stein fillings, rational surface singularities, and plane curve arrangements
针对约化基本闭链的有理曲面奇点，研究构建辛几何类比构造，通过辛圆盘配置描述其链环的所有Stein填充，证明部分奇点存在非Milnor纤维的Stein填充，同时给出极小消解自交数≤-5时Stein填充唯一的结论。
We introduce a new class of line arrangements in the projective plane, called nearly supersolvable, and show that any arrangement in this class is either free or nearly free. More precisely, we show that the minimal degree of a Jacobian syzygy for the defining equation of the line arrangement, which is a subtle algebraic invariant, is determined in this case by the combinatorics. When such a line arrangement is nearly free, we discuss the splitting types and the jumping lines of the associated rank two vector bundle, as well as the corresponding jumping points, introduced recently by S. Marche…
In a recent paper by Cook, et al., which introduced the concept of unexpected plane curves, the focus was on understanding the geometry of the curves themselves. Here we expand the definition to hypersurfaces of any dimension and, using constructions which appeal to algebra, geometry, representation theory and computation, we obtain a coarse but complete classification of unexpected hypersurfaces. In particular, we determine each $(n,d,m)$ for which there is some finite set of points $Z\subset\mathbb P^n$ with an unexpected hypersurface of degree $d$ in $\mathbb P^n$ having a general point $P$…
On homological properties of conic-line arrangements with simple singularities
研究复射影平面中光滑圆锥和线的排列，其奇点限于节点、切点和普通三重点；第一部分给出缺陷ν(C)=3的加一生成圆锥排列的数值限制，第二部分关注总次数≤6的排列并识别Ziegler对。
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…
The motivating problem addressed by this paper is to describe those non-degenerate sets of points $Z$ in $\mathbb P^3$ whose general projection to a general plane is a complete intersection of curves in that plane. One large class of such $Z$ is what we call $(m,n)$-grids. We relate this problem to the {\em unexpected cone property} ${\mathcal C}(d)$, a special case of the unexpected hypersurfaces which have been the focus of much recent research. After an analysis of ${\mathcal C}(d)$ for small $d$, we show that a non-degenerate set of $9$ points has a general projection that is the complete …
Complexes inattendus de droites de saut (Unexpected complex of jumping lines)
本文研究P₃上秩2向量丛E的跳跃线相关复形，证明了若P₃∨中约化不可约曲线C上所有平面H对应的E_H不稳定，则C为直线；并建立了L∨⊂W(E)与E截面零簇包含L高阶无穷小邻域的等价条件。
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