A tight bound on the projective dimension of four quadrics

Motivated by Stillman's question, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring is at most 6; moreover, this bound is tight. We achieve this bound, in part, by giving a characterization of the low degree generators of ideals primary to height three primes of multiplicities one and two.     

Essential dimension of Hermitian spaces

Given an hermitian space we compute its essential dimension, Chow motive and prove its incompressibility in certain dimensions.     

Essential dimension: a survey

In the paper we survey research on the essential dimension. The highlights of the survey are the computations of the essential dimensions of finite groups, groups of multiplicative type and the spinor groups. We present self-contained proofs of these cases and give applications in the theory of simple algebras and quadratic forms.     

Essential dimension of involutions and subalgebras

Essential dimension is an invariant of algebraic groups G over a field F that measures the complexity of G-torsors over field extensions of F. We use theorems of N. Karpenko about the incompressibility of Severi-Brauer varieties and quadratic Weil transfers of Severi-Brauer varieties to compute the essential dimension of some closed subgroups of R _K/F (GL ₁(A)), where A is a central division K-algebra of prime power degree and K/F is a separable field extension of degree ≤ 2. In particular, we determine the essential dimension of the group Sim(A, σ) of similitudes of (A, σ), where σ is an F-involution on A, and the essential dimension of the normalizer $N_{GL_1 (A)} \left( {GL_1 \left( B \right)} \right)$ , where B is a separable subalgebra of A.     

Essential dimension of central simple algebras

Let p be a prime integer, 1≤s≤r be integers and F be a field of characteristic different from p. We find upper and lower bounds for the essential p-dimension ed _p (\( Al{{g}_{{{{p}^r},{{p}^s}}}} \)) of the class \( Al{{g}_{{{{p}^r},{{p}^s}}}} \) of central simple algebras of degree p ^r and exponent dividing p ^s . In particular, we show that ed(Alg _8,2)=ed₂(Alg _8,2)=8 and ed _p (\( Al{{g}_{{{{p}^2},p}}} \))=p ²+p for p odd.     

Essential dimension of Abelian varieties over number fields

We show that the essential dimension of a non-trivial Abelian variety over a number field is infinite. To cite this article: P. Brosnan, R. Sreekantan, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Essential dimension of finite groups in prime characteristic

Let F be a field of characteristic $p>0$ and G be a smooth finite algebraic group over F. We compute the essential dimension ${\mathrm{ed}}_F(G;p)$ of G at p. That is, we show that

${\mathrm{ed}}_F(G;p)=\{\begin{array}{cc}1,\hfill & \text{if}p\text{divides}|G|,\text{and}\hfill \\ 0,\hfill & \text{otherwise.}\hfill \end{array}$

     

Essential domains and two conjectures in dimension theory

This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings in the context of (locally) essential domains.

     

Essential dimension of algebraic groups,including bad characteristic

Let A and G be finite groups of relatively prime orders and assume that A acts on G via automorphisms. We study how certain conditions on G imply its solvability when we assume the existence of a unique A-invariant Sylow p-subgroup for p equal to 2 or 3.     

About C-rigid quadrics

In the article strongly nondegenerate (k, n)-quadrics all of whose linear automorphisms are of the formzz, ¦¦2, {0} are considered. Quadrics all of whose linear automorphisms are of this form were calledc-rigid by V. Beloshapka. The main result of the article is the following: anyc-rigid strongly nondegenerate (k, n)-quadric has no nonlinear automorphisms. A table indicating the relationship between linear and nonlinear automorphisms for (k, n)-quadrics is presented.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 224–229, February, 1996.     

Categorical characterization of quadrics

We give a characterization of smooth quadrics in terms of the existence of full exceptional collections of certain type, which generalizes a result of C.Vial for projective spaces.     

Essential extensions, excellent extensions and finitely presented dimension

In Section 1 of this paper, we investigate the finitely presented dimension of an essential extension for a module, and obtain results concerning an essential extension of a torsion-free module. We partially answer the question: When is an essential extension of a finitely presented module (an almost finitely presented module) also finitely presented (almost finitely presented)? In Section 2, we study theC-excellent extensions and the finitely presented dimensions. We obtain some results on the homological dimensions of matrix rings and group rings.     

Essential dimension and the second serre conjecture for exceptional groups

We study the essential dimension of exceptional connected simply connected algebraic groups over algebraically closed fields. In the present paper, we find upper estimates for the essential dimensions of the groupsF ₄ ,E ₆ , andE ₇ . For the groupF ₄ , the upper estimate, thus obtained coincides with the known lower estimate. We also prove the second Serre conjecture for the groupE ₆ and for a function field. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 539–547, October, 2000.     

Real classification of complex quadrics

Canonical forms are given for complex quadric surfaces (and conics) under real changes of variable. The basic idea is to use known results on pairs of real quadratic forms.