Publication: Linear free Divisors and quiver representation
Abstract
OZET Bu tezin amacı, cebirsel geometrinin ¨onemli bir kavramı olan Lineer Serbest B¨olenleri (Linear Free Divisors) incelemek ve bu kavramın C¸ izgelerin Temsil Teorisi (Quiver Representation) ile arasındaki ilis¸kisini anlamaktır. Bir polinomun sıfır k¨umesi olarak tanımlanan divisor kavramının ¨ozel bir durumu olan ve smooth (non-singular) bir y¨uzeyin sıfır k¨umesi olan free divisor kavramı ilk olarak Kyoji Saito tarafından ortaya atılmıs¸tır. Bir logarithmic vekt¨or alanı, her noktayı uygun bir tanjant vekt¨or¨une kars¸ılık getiren ve ¨ozel olarak etkiles¸ime girdi˘gi her analitik fonksiyonu, bu fonksiyonun idealine eleman yapan ¨ozel bir vekt¨or alanıdır. Bu sıfır k¨umelerinin (divisor) serbest olması, bu k¨umenin her noktasındaki logarithmic vect¨or alanları mod¨ul¨un¨un (DerCn;p(log D)) bir serbest OCn mod¨ul olmasıyla, yani bir tabanının bulunabilmesiyle, lineer olması ise DerCn;p(log D) ic¸in bulunan bir bazın katsayılarının (ai j ler) lineer fonksiyon olması ile ac¸ıklanır. Kyoji Saito’ nun bunu ac¸ıklayan ve bu tezde de kullanılan o¨nemli bir kriteri vardır [10]. O¨ te yandan bir kısım o¨zel c¸izgeler olan Dynkin tipi c¸izgeler (An; Dn E6; E7; E8) incelenmis¸tir. Bu c¸izgelerin sonlu temsillerinin oldu˘gu Peter Gabriel tarafından g¨osterilmis¸tir [9]. Bu durum bu c¸izgelerin g¨osterilis¸ diskriminantının reduced (katlı k¨ok ic¸ermeme) olmasını sa˘glar. E˘ger ek olarak bu c¸izgelerin boyut vekt¨or¨u real root (q(d)=1) ise bu matrisin determinantı bir lineer serbest b¨olen (linear free divisor) tanımlar
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), i = Xn j=1 ai j @ @xj 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. i
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), i = Xn j=1 ai j @ @xj 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. i