Commutative ring theory: Regular rings

Proofs from THE BOOK, by Martin Aigner and Giinter M. Ziegler. Pp.199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag). I would like to begin with the authors' words from the Preface of this book: Paul Erd6s liked to talk about The Book, in which God maintains the perfect proofs for mathematical theorems, following the dictum of G. H. Hardy that there is no permanent place for ugly mathematics. ErdSs also said that 'you need not believe in God but, as a mathematician, you should believe in The Book'. This excellent book is dedicated to the memory of Paul ErdSs. Paul himself wrote several pages but he is not listed as a co-author because of his death in the summer of 1996. The idea of the book is to present beautiful proofs of some of the most exciting and historically important mathematical theorems. The book focuses on human endeavour to make the proofs more and more simple and striking.

Proofs from THE BOOK , by Martin Aigner and Günter M. Ziegler. Pp. 199. £19. 1999. ISBN 3 540 63698 6 (Springer-Verlag).

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Arrangements Of Hyperplanes

Star configurations in Pn

Algebraic geometric codes: basic notions

This work deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both weighted homogeneous and homogeneous polynomials, allowing to introduce new families of free divisors not coming from either hyperplane arrangements or discriminants in singularity theory.

Homology of homogeneous divisors

One deals with arbitrary reduced free divisors in a polynomial ring over a field of characteristic zero, by stressing the ideal theoretic and homological behavior of the corresponding singular locus. A particular emphasis is given to both weighted homogeneous and homogeneous polynomials, allowing to introduce new families of free divisors which do not come from hyperplane arrangements nor as explicit discriminants from singularity theory.

Homology of Homogeneous Divisors

The Orlik-Solomon algebra is the cohomology ring of the complement of a hyperplane arrangement $\A \subseteq \C^n$; it is the quotient of an exterior algebra $\Lambda(V)$ on $|\A|$ generators. In \cite{ot1}, Orlik and Terao introduced a commutative analog $Sym(V^*)/I$ of the Orlik-Solomon algebra to answer a question of Aomoto and showed the Hilbert series depends only on the intersection lattice $L(\A)$. In \cite{fr}, Falk and Randell define the property of 2-formality; in this note we study the relation between 2-formality and the Orlik-Terao algebra. Our main result is a necessary and sufficient condition for 2-formality in terms of the quadratic component $I_2$ of the Orlik-Terao ideal $I$. The key is that 2-formality is determined by the tangent space $T_p(V(I_2))$ at a generic point $p$.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2009/0016/0001/MRL-2009-0016-0001-a017.pdf

The Orlik-Terao algebra and 2-formality

It is shown that the Orlik-Terao algebra is graded isomorphic to the special fiber of the ideal $I$ generated by the $(n-1)$-fold products of the members of a central arrangement of size $n$. This momentum is carried over to the Rees algebra (blowup) of $I$ and it is shown that this algebra is of fiber-type and Cohen-Macaulay. It follows by a result of Simis-Vasconcelos that the special fiber of $I$ is Cohen-Macaulay, thus giving another proof of a result of Proudfoot-Speyer about the Cohen-Macauleyness of the Orlik-Terao algebra.

https://msp.org/ant/2018/12-6/ant-v12-n6-p02-s.pdf

A blowup algebra for hyperplane arrangements

We study the structure of the Rees algebra of almost complete intersection ideals of finite colength in low-dimensional polynomial rings over fields. The main tool is a mix of Sylvester forms and iterative mapping cone construction. The material developed spins around ideals of forms in two or three variables in the search of those classes for which the corresponding Rees ideal is generated by Sylvester forms and is almost Cohen–Macaulay. A main offshoot is in the case where the forms are monomials. Another consequence is a proof that the Rees ideals of the base ideals of certain plane Cremona maps (e.g., de Jonquieres maps) are generated by Sylvester forms and are almost Cohen–Macaulay.

Stefan O. Tohaneanu

Papers

Homology of homogeneous divisors

Homology of Homogeneous Divisors

The Orlik-Terao algebra and 2-formality

A blowup algebra for hyperplane arrangements

The ubiquity of Sylvester forms in almost complete intersections