Linear free divisors and quiver representation


Thesis Type: Postgraduate

Institution Of The Thesis: Marmara University, Institute for Graduate Studies in Pure and Applied Sciences, Turkey

Approval Date: 2014

Thesis Language: English

Student: ABUZER GÜNDÜZ

Principal Supervisor (For Co-Supervisor Theses): Ünsal Tekir

Abstract:

The aim of this thesis is to study linear free divisors in algebraic geometry. An hypersurface D = V(h) 2 Cn is called a divisor. A vector field over Cn is called a logarithmic vector field (or derivation) if p(h) (h) for any smooth point p of D. The sheaf of logarithmic derivations is denoted by DerCn;p(log D) = f 2 Der(Cn)g : (h) (h)g We say that D is free if Der(log D) is a locally free OCn;p module. We use the following criterion to determine when a divisor is free. Theorem 0.0.1. (Saito's Criterion) [2] The OCn;p module DerCn;p(log D) is free if and only if there exist n elements 1; : : : ; n in DerCn;p(log D) (i.e vector fields), in such that det(ai j(x)) is equal to h up to an invertible factor where ai j 2 OCn;p which are coecients of 1; : : : ; n; i = 1; : : : ; n: The vector fields 1; : : : ; n form a basis of DerCn;p(log D): On the other hand, a quiver Q is a Dynkin quiver if its underlying graph is a Dynkin diagram of type An; Dn; E6; E7 or E8: Gabriel proved in [9] that the Dynkin quivers are precisely those of "finite representation type". This guarantees that the discriminant D in Rep(Q; d) of Q quiver is always reduced. In addition to, if the dimension vector d is a root, then the determinant of discriminant  defines a linear free divisor [2]. We explain how linear free divisors arise as discriminants in quiver representations.