An algorithm for a lifted Massey triple product of a smooth projective plane curve

2020 ◽  
Vol 30 (08) ◽  
pp. 1651-1669
Author(s):  
Younggi Lee ◽  
Jeehoon Park ◽  
Junyeong Park ◽  
Jaehyun Yim
Keyword(s):  
Projective Plane ◽  
Homogeneous Polynomial ◽  
Plane Curve ◽  
Main Idea ◽  
Triple Product ◽  
Cup Product ◽  
Smooth Projective Plane ◽  
Defining System ◽  
Polynomial Formula ◽  
Massey Triple Product

We provide an explicit algorithm to compute a lifted Massey triple product relative to a defining system for a smooth projective plane curve [Formula: see text] defined by a homogeneous polynomial [Formula: see text] over a field. The main idea is to use the description (due to Carlson and Griffiths) of the cup product for [Formula: see text] in terms of the multiplications inside the Jacobian ring of [Formula: see text] and the Cech–deRham complex of [Formula: see text]. Our algorithm gives a criterion whether a lifted Massey triple product vanishes or not in [Formula: see text] under a particular nontrivial defining system of the Massey triple product and thus can be viewed as a generalization of the vanishing criterion of the cup product in [Formula: see text] of Carlson and Griffiths. Based on our algorithm, we provide explicit numerical examples by running the computer program.

ON FINITE BRANCHED UNIFORMIZATIONS OF THE PROJECTIVE PLANE

2013 ◽  
Vol 24 (02) ◽  
pp. 1350017
Author(s):  
A. MUHAMMED ULUDAĞ ◽  
CELAL CEM SARIOĞLU
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Free Product ◽  
Plane Curve ◽  
Complex Projective Plane ◽  
Galois Covering ◽  
Cyclic Groups ◽  
Galois Coverings ◽  
Complex Projective ◽  
Branching Index

We give a brief survey of the so-called Fenchel's problem for the projective plane, that is the problem of existence of finite Galois coverings of the complex projective plane branched along a given divisor and prove the following result: Let p, q be two integers greater than 1 and C be an irreducible plane curve. If there is a surjection of the fundamental group of the complement of C into a free product of cyclic groups of orders p and q, then there is a finite Galois covering of the projective plane branched along C with any given branching index.

scholarly journals Symplectic mapping class group relations generalizing the chain relation

2016 ◽  
Vol 27 (12) ◽  
pp. 1650096 ◽  
Author(s):  
Bahar Acu ◽  
Russell Avdek
Keyword(s):  
Mapping Class Group ◽  
Homogeneous Polynomial ◽  
Class Group ◽  
Symplectic Manifolds ◽  
Mapping Class ◽  
Symplectic Mapping ◽  
Group Relations ◽  
Type Boundary ◽  
Polynomial Formula ◽  
Classical Chain

In this paper, we examine mapping class group relations of some symplectic manifolds. For each [Formula: see text] and [Formula: see text], we show that the [Formula: see text]-dimensional Weinstein domain [Formula: see text], determined by the degree [Formula: see text] homogeneous polynomial [Formula: see text], has a Boothby–Wang type boundary and a right-handed fibered Dehn twist along the boundary that is symplectically isotopic to a product of right-handed Dehn twists along Lagrangian spheres. We also present explicit descriptions of the symplectomorphisms in the case [Formula: see text] recovering the classical chain relation for the torus with two boundary components.

scholarly journals ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES

2005 ◽  
Vol 92 (1) ◽  
pp. 99-138 ◽  
Author(s):  
J. FERNÁNDEZ DE BOBADILLA ◽  
I. LUENGO-VELASCO ◽  
A. MELLE-HERNÁNDEZ ◽  
A. NÉMETHI
Keyword(s):  
Singular Point ◽  
Projective Plane ◽  
Plane Curve ◽  
Topological Type ◽  
Homology Sphere ◽  
Plane Curves ◽  
Curve Singularity ◽  
Witten Invariant ◽  
Rational Homology ◽  
Rational Homology Sphere

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

scholarly journals Hirzebruch-Type Inequalities Viewed as Tools in Combinatorics

10.37236/8335 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Piotr Pokora
Keyword(s):  
Projective Plane ◽  
Plane Curve ◽  
Combinatorial Problems ◽  
Complex Projective Plane ◽  
Open Problems ◽  
Line Arrangements ◽  
Dirac Conjecture ◽  
Complex Projective

The main purpose of this survey is to provide an introduction, algebro-topological in nature, to Hirzebuch-type inequalities for plane curve arrangements in the complex projective plane. These inequalities gain more and more interest due to their utility in many combinatorial problems related to point or line arrangements in the plane. We would like to present a summary of the technicalities and also some recent applications, for instance in the context of the Weak Dirac Conjecture. We also advertise some open problems and questions.

Weierstrass points of order three on smooth quartic curves

2019 ◽  
Vol 18 (01) ◽  
pp. 1950020 ◽  
Author(s):  
Alwaleed Kamel ◽  
Waleed Khaled Elshareef
Keyword(s):  
Projective Plane ◽  
Weierstrass Points ◽  
Smooth Projective Plane ◽  
Quartic Curve ◽  
The Subject ◽  
Quartic Curves

In this paper, we study the [Formula: see text]-Weierstrass points on smooth projective plane quartic curves and investigate their geometry. Moreover, we use a technique to determine in a very precise way the distribution of such points on any smooth projective plane quartic curve. We also give a variety of examples that illustrate and enrich the subject.

scholarly journals SEVERI VARIETIES AND BRANCH CURVES OF ABELIAN SURFACES OF TYPE (1, 3)

2002 ◽  
Vol 13 (03) ◽  
pp. 227-244 ◽  
Author(s):  
H. LANGE ◽  
E. SERNESI
Keyword(s):  
Linear System ◽  
Plane Curve ◽  
Main Idea ◽  
Abelian Surface ◽  
Severi Varieties ◽  
Severi Variety ◽  
System L ◽  
Discriminant Curve ◽  
Branch Curve

A polarized abelian surface (A, L) of type (1, 3) induces a 6:1 covering of A onto the projective plane with branch curve, a plane curve B of degree 18. The main result of the paper is that for a general abelian surface of type (1, 3), the curve B is irreducible and reduced and admits 72 cusps, 36 nodes or tacnodes, each tacnode counting as 2 nodes, 72 flexes and 36 bitangents. The main idea of the proof is that for a general (A, L) the discriminant curve in the linear system |L| coincides with the closure of the Severi variety of curves in |L| admitting a node and is dual to the curve B in the sense of projective geometry.

On the irreducibility of Hilbert scheme of surfaces of minimal degree

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Fedor Bogomolov ◽  
Viktor Kulikov
Keyword(s):  
Projective Plane ◽  
Hilbert Scheme ◽  
Special Class ◽  
Main Idea ◽  
Rational Curves ◽  
Minimal Degree ◽  
Irreducible Components

AbstractThe article contains a new proof that the Hilbert scheme of irreducible surfaces of degree m in ℙm+1 is irreducible except m = 4. In the case m = 4 the Hilbert scheme consists of two irreducible components explicitly described in the article. The main idea of our approach is to use the proof of Chisini conjecture [Kulikov Vik.S., On Chisini’s conjecture II, Izv. Math., 2008, 72(5), 901–913 (in Russian)] for coverings of projective plane branched in a special class of rational curves.

Characterizing some polarized Fano fibrations via Hilbert curves

2020 ◽  
pp. 2250046
Author(s):  
Antonio Lanteri ◽  
Andrea Luigi Tironi
Keyword(s):  
Plane Curve ◽  
Hilbert Curve ◽  
Special Shape ◽  
Fano Manifolds ◽  
Adjunction Theory ◽  
Adjoint Bundle ◽  
Fano Fibrations ◽  
Polynomial Formula ◽  
Low Dimensional ◽  
Hilbert Curves

The Hilbert curve of a complex polarized manifold [Formula: see text] is the complex affine plane curve of degree [Formula: see text] defined by the Hilbert-like polynomial [Formula: see text], where [Formula: see text] is the canonical bundle of [Formula: see text] and [Formula: see text] and [Formula: see text] are regarded as complex variables. A natural expectation is that this curve encodes several properties of the pair [Formula: see text]. In particular, the existence of a fibration of [Formula: see text] over a variety of smaller dimension induced by a suitable adjoint bundle to [Formula: see text] translates into the fact that the Hilbert curve has a quite special shape. Along this line, Hilbert curves of special varieties like Fano manifolds with low coindex, as well as fibrations over low-dimensional varieties having such a manifold as general fiber, endowed with appropriate polarizations, are investigated. In particular, several polarized manifolds relevant for adjunction theory are completely characterized in terms of their Hilbert curves.

scholarly journals Hypersurfaces with linear type singular loci

2019 ◽  
Vol 19 (09) ◽  
pp. 2050169
Author(s):  
Amir Behzad Farrahy ◽  
Abbas Nasrollah Nejad
Keyword(s):  
Projective Plane ◽  
Plane Curve ◽  
Isolated Singularity ◽  
Linear Type ◽  
Jacobian Ideal ◽  
Quartic Curve ◽  
Singular Loci ◽  
Necessary And Sufficient Criteria ◽  
Simple Singularities ◽  
Necessary And Sufficient

In this paper, necessary and sufficient criteria for the Jacobian ideal of a reduced hypersurface with isolated singularity to be of linear type are presented. We prove that the gradient ideal of a reduced projective plane curve with simple singularities ([Formula: see text]) is of linear type. We show that any reduced projective quartic curve is of gradient linear type.

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