Syzygies of Toric Varieties (draft) This Is a Chapter from a Book

1. Basics Let A = fa 1 ; : : :; a n g be a subset of N d n f0g, A be the matrix with columns a i , and suppose that rank(A) = d. Consider the polynomial ring S = kx 1 x i 7 ! t a i = t a i1 1 : : : t a id d is called a toric ideal. For an integer vector v = (v 1 ; : : :; v n) we set x v = x v 1 1 : : : x v n n. In this notation, the map ' acts as x v 7 ! t Av : Deenition 1.1. The toric ring is S=I A = kt a 1 ; : : :; t a n ] = NA; where the second isomorphism is given by t a 7 ! a. A toric variety is a variety parametrized by nitely many monomials. Such varieties are extremely useful in illustrating phenomena in algebraic geometry. Our deenition is weaker than the deenition used in some classical texts: in Fulton, Danilov] S=I A is called toric if the points fa 1 ; : : :; a n g form the monoid basis of N d \ for some rational cone. A toric ring in our sense is often called a semigroup ring. A toric ring in the sense of Fulton, Danilov] is normal; whereas 1