scholarly journals Line arrangements and configurations of points with an unexpected geometric property

2018 ◽  
Vol 154 (10) ◽  
pp. 2150-2194 ◽  
Author(s):  
D. Cook ◽  
B. Harbourne ◽  
J. Migliore ◽  
U. Nagel
Keyword(s):  
Rational Curves ◽  
Fixed Degree ◽  
Fat Points ◽  
Line Arrangements ◽  
Base Points ◽  
Complete Linear System ◽  
Lefschetz Property ◽  
Linear Subsystem ◽  
Strong Lefschetz Property ◽  
Special Case

We propose here a generalization of the problem addressed by the SHGH conjecture. The SHGH conjecture posits a solution to the question of how many conditions a general union$X$of fat points imposes on the complete linear system of curves in$\mathbb{P}^{2}$of fixed degree$d$, in terms of the occurrence of certain rational curves in the base locus of the linear subsystem defined by$X$. As a first step towards a new theory, we show that rational curves play a similar role in a special case of a generalized problem, which asks how many conditions are imposed by a general union of fat points on linear subsystems defined by imposed base points. Moreover, motivated by work of Di Gennaro, Ilardi and Vallès and of Faenzi and Vallès, we relate our results to the failure of a strong Lefschetz property, and we give a Lefschetz-like criterion for Terao’s conjecture on the freeness of line arrangements.

scholarly journals Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

2020 ◽  
Vol 2020 (21) ◽  
pp. 8139-8182 ◽  
Author(s):  
Jarosław Buczyński ◽  
Nathan Ilten ◽  
Emanuele Ventura
Keyword(s):  
Linear System ◽  
Smooth Curve ◽  
Rational Curves ◽  
Arithmetic Genus ◽  
Smooth Curves ◽  
Low Degree ◽  
Complete Linear System ◽  
Osculating Spaces ◽  
Special Case ◽  
Schubert Cycles

Abstract In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree $d$ rational curves in $\mathbb{P}^n$ when $d-n\leq 3$ and $d<2n$. Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption $d<2n$, the arithmetic genus of any non-degenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d-n$.

The Strong Lefschetz Property and the Schur–Weyl Duality

2013 ◽  
pp. 211-234
Author(s):  
Tadahito Harima ◽  
Toshiaki Maeno ◽  
Hideaki Morita ◽  
Yasuhide Numata ◽  
Akihito Wachi ◽  
...  
Keyword(s):  
Weyl Duality ◽  
Lefschetz Property ◽  
Strong Lefschetz Property

Cohomology Rings and the Strong Lefschetz Property

2013 ◽  
pp. 189-199
Author(s):  
Tadahito Harima ◽  
Toshiaki Maeno ◽  
Hideaki Morita ◽  
Yasuhide Numata ◽  
Akihito Wachi ◽  
...  
Keyword(s):  
Cohomology Rings ◽  
Lefschetz Property ◽  
Strong Lefschetz Property

On the canonical systems of certain algebraic surfaces

1937 ◽  
Vol 33 (3) ◽  
pp. 311-314
Author(s):  
D. Pedoe
Keyword(s):  
Algebraic Surface ◽  
Base Point ◽  
Canonical System ◽  
Algebraic Surfaces ◽  
Simple Point ◽  
Canonical Systems ◽  
Base Points ◽  
Complete Linear System ◽  
Pass Through ◽  
Simple Points

A complete linear system of curves on an algebraic surface may have assigned base points. The canonical system, from its definition, has no assigned base points at simple points of the surface. But we may construct surfaces on which, all the same, the canonical system has “accidental base points” at simple points of the surface. The classical example, due to Castelnuovo, is a quintic surface with two tacnodes. On this surface the canonical system is cut out by the planes passing through the two tacnodes. These planes also pass through the simple point in which the join of the two tacnodes meets the surface again. This point is the accidental base point of the canonical system on the quintic surface.

scholarly journals The Geometry of the Weak Lefschetz Property and Level Sets of Points

2008 ◽  
Vol 60 (2) ◽  
pp. 391-411 ◽  
Author(s):  
Juan C. Migliore
Keyword(s):  
Level Set ◽  
Level Sets ◽  
Hilbert Function ◽  
Worst Case ◽  
Weak Lefschetz Property ◽  
Lefschetz Property ◽  
Strong Lefschetz Property ◽  
Set Of Points

AbstractIn a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.

scholarly journals On the formality and strong Lefschetz property of symplectic manifolds – Erratum

2009 ◽  
Vol 147 (1) ◽  
pp. 255-255
Author(s):  
Taek Kyu Hwang ◽  
Jin Hong Kim
Keyword(s):  
Fundamental Group ◽  
Symplectic Manifold ◽  
Symplectic Manifolds ◽  
Finite Covering ◽  
Massey Product ◽  
Lefschetz Property ◽  
The Stability ◽  
Simply Connected ◽  
Open Question ◽  
Strong Lefschetz Property

Professor Vicente Muñoz kindly informed us that there is an inaccuracy in Lemma 3.5 of [1]. The correct statement of Lemma 3.5 is now that the fundamental group π1(X′) of the manifold X′ is Z, since the monodromy coming from φ8 does not imply that g4 = g4−1. Therefore, what we have actually constructed in Section 3 of [1] is a closed non-formal 8-dimensional symplectic manifold with π1 = Z whose triple Massey product is non-zero, so that the simply-connectedness in Theorem 1.1 should be dropped. As far as we know, the existence of a simply connected closed non-formal 8-dimensional symplectic manifold whose triple Massey product is non-zero still remains an open question. All other main results, especially Theorem 1.2 and Corollary 1.3, in [1] are not affected by this mistake. Furthermore, the stability of the non-formality under a finite covering as in Subsection 3.3 holds in general. We want to thank Professor Muñoz for his careful reading.

scholarly journals Homaloidal nets and ideals of fat points I

2016 ◽  
Vol 19 (1) ◽  
pp. 54-77 ◽  
Author(s):  
Zaqueu Ramos ◽  
Aron Simis
Keyword(s):  
Linear System ◽  
Fat Points ◽  
Base Points ◽  
Algebraic Properties ◽  
The Ideal

We consider plane Cremona maps with proper base points and the base ideal generated by the linear system of forms defining the map. The object of this work is to study the link between the algebraic properties of the base ideal and those of the ideal of these points fattened by the virtual multiplicities arising from the linear system. We reveal conditions which naturally regulate this association, with particular emphasis on the homological side. While most classical numerical inequalities concern the three highest virtual multiplicities, here we emphasize also the role of one single highest multiplicity. In this vein we describe classes of Cremona maps for large and small values of the highest virtual multiplicity. We also deal with the delicate question as to when is the base ideal non-saturated and consider the structure of its saturation.

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