An algebraic approach to finite projective planes

Advances in Geometry, 2000

We establish the connections between finite projective planes admitting a collineation group of Lenz-Barlotti type I.3 or I.4, partially transitive planes of type (3) in the sense of Hughes, and planes admitting a quasiregular collineation group of type (g) in the Dembowski-Piper classification; our main tool is an equivalent description by a certain type of di¤erence set relative to disjoint subgroups which we will call a neo-di¤erence set. We then discuss geometric properties and restrictions for the existence of planes of Lenz-Barlotti class I.4. As a side result, we also obtain a new synthetic description of projective triangles in desarguesian planes.

Proceedings of The American Mathematical Society - PROC AMER MATH SOC, 1995

We construct an infinite projective plane with Lenz-Barlotti class I. Moreover, the plane is almost strongly minimal in a very strong sense: each automorphism of each line extends uniquely to an automorphism of the plane.

Archiv der Mathematik, 1974

Compositio Mathematica - COMPOS MATH, 2000

The paper provides a combinatorial method to decide when the space of local systems with nonvanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of Orlik–Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg–Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components.

Cornell University - arXiv, 2021

Let ∆ be an (abstract) simplicial complex on n vertices. One can define the Artinian monomial algebra A(∆) = k[x 1 ,. .. , x n ]/ x 2 1 ,. .. , x 2 n , I ∆ , where k is a field of characteristic 0 and I ∆ is the Stanley-Reisner ideal associated to ∆. In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of A(∆) in terms of the simplicial complex ∆. We are able to completely analyze when WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck in [22]. We give a complete characterization of all 2dimensional pseudomanifolds ∆ such that A(∆) satisfies WLP. We also construct Artinian Gorenstein algebras that fail WLP by combining our results and the standard technique of Nagata idealization.

Manuscripta Mathematica, 1994

In this paper we study graded Betti numbers of projective varieties. Using a spectral sequence argument, we establish an algebraic version of a duality Theorem proved first by Mark Green. Our approach doesn't require any smoothness or characteristic 0 assumption. We then study the graded Betti numbers of finite subschemes of a rational normal curve and apply these results to generalize another theorem of Mark Green, the Kp.1 theorem, to some non-reduced schemes. Our result applies for instance in the case of ribbons.

Combinatorica, 2007

Let Π be a projective plane of order n in Lenz-Barlotti class I.4, and assume that n is a multiple of 3. Then either n = 3 or n is a multiple of 9.

Journal of Pure and Applied Algebra, 1998

Associated to any simplicial complex A on n vertices is a square-free monomial ideal I.1 in the polynomial ring A = k[x~, ,x,1, and its quotient k[A] = A/IA known as the Stanley-Reisner ring. This note considers a simplicial complex A* which is in a sense a canonical Alexander dual to A, previously considered in [I, 51. Using Alexander duality and a result of Hochster computing the Betti numbers dimkTor,A(k[d],k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in A*. As corollaries, we prove that 1~ has a linear resolution as A-module if and only if A* is Cohen-Macaulay over k. and show how to compute the Betti numbers dimkTort(k[A],k)

2007

We determine an explicit presentation by generators and relations of the cohomology algebra $H^*(\mathbb P^2\setminus C,\mathbb C)$ of the complement to an algebraic curve $C$ in the complex projective plane $\mathbb P^2$, via the study of log-resolution logarithmic forms on $\mathbb P^2$. As a first consequence, we derive that $H^*(\mathbb P^2\setminus C,\mathbb C)$ depends only on the following finite pieces of data: the number of irreducible components of $C$ together with their degrees and genera, the number of local branches of each component at each singular point, and the intersection numbers of every two distinct local branches at each singular point of $C$. This finite set of data is referred to as the weak combinatorial type of $C$. A further corollary is that the twisted cohomology jumping loci of $H^*(\mathbb P^2\setminus C,\mathbb C)$ containing the trivial character also depend on the weak combinatorial type of $C$. Finally, the explicit construction of the generators and relations allows us to prove that complements of plane projective curves are formal spaces in the sense of Sullivan.

International Journal of Scientific and Research Publications (IJSRP)

In this paper we study the excision property of short sequence to get the long sequence of the dihedral homology and reflexive homology of polynomial algebra. We give a new application of these theorems if we take a new category of graded lie algebras. Another application is relative homology.