Mimar Sinan Fine Arts University General Seminars

Pencils of conics over finite fields
Michel Lavrauw
Sabancı Üniversitesi, Turkey
Özet : A pencil of quadrics is a two-dimensional subspace of the vector space of quadrics of a given projective space, and is therefore defined by a pair of quadratic forms on a given vector space. The study of pencils of quadrics dates back to the late 19th century when Weierstrass (1868) studied the irrational form of a regular pencil and Kronecker (1890) discussed the singular case (both over the complex numbers). The methods developed by Weierstrass and Kronecker resulted in the description of pencils of quadrics using the theory of invariant factors and elementary divisors. Dickson was the first to study pencils over finite fields (for which the aforementioned method fails). In 1908 Dickson classified pencils of quadrics in projective planes (i.e. conics) over finite fields of odd characteristic, giving explicit coordinate transformations in order to reduce the families of ternary quadratic forms to canonical representatives of the associated equivalence classes of pencils. Part of his proof relies on the knowledge of the number of irreducible cubics of a given form and refers to Dickson's treatise on Linear Groups. Recently, together with Tomasz Popiel, we completed the classification of pencils of conics over finite fields of even characteristic. In this talk I will explain how this classification follows from our work on the symmetric representation of orbits of lines (under the natural action of GL(3,K)xGL(3,K)) in the 2-fold tensor product of 3-dimensional vector spaces. Our proof is mostly geometric and works for finite fields of both even and odd characteristic, and so in particular we recover Dickson's classification. The proof also works for the reals and all algebraically closed fields (the approach using elementary divisors also fails for algebraically closed fields of characteristic 2). Particularly interesting is the case of pencils without degenerate conics (a case which does not appear if the field is algebraically closed). Our results imply that each two such pencils are projectively equivalent.
 Tarih : 01.11.2018 Saat : 16:00 Yer : MSGSÜ Matematik Bölümü Seminer Salonu Dil : English