Reading course report , on the book Ideals , Varieties and Algorithms

This report gives an overview of the main ideas in Chapter 9 of the studied book, about the dimension of a variety. After recalling some definitions and basic properties about projective varieties, homogeneous ideals and Gröbner bases, we will begin our study of the dimension of a variety with the observation of a geometric way to define the dimension of a variety defined by a monomial ideal. Then, we show how to characterize this dimension by algebraic properties of the defining ideal. Using Gröbner bases to link the ideal of a variety to the monomial ideal of its leading terms, this allows us to give a satisfying definition to the dimension of any projective variety; we conclude with some properties linked to the dimension of a variety. 1 Ideals, varieties and Gröbner bases In this report we will deal with projective varieties, that is, varieties in a projective space that are defined by homogeneous ideals of the ring of polynomials k[x0, . . . , xn], where k is an infinite field and n > 1. In this section, we briefly recall how one can define these two central objects — ideals and varieties — and how they are linked; we also give basic properties of Gröbner bases, which allow to effectively compute objects and effectively solve problems related to ideals and varieties. 1.1 Homogeneous ideals and projective varieties Two points (a0, . . . , an) and (a ′ 0, . . . , a ′ n) in the affine space k n+1 are said to be equivalent if (a0, . . . , an) = (λa ′ 0, . . . , λa ′ n) for some nonzero λ ∈ k; this defines an equivalence relation ∼ over kn+1 − {0}. In the projective space Pn(k), we work modulo ∼, which means that two equivalent points will be considered equal. Definition 1. The n-th dimensional projective space over k is the set of equivalence classes Pn(k) = (kn+1 − {0})/ ∼. Thus each nonzero (a0, . . . , an) ∈ kn+1 defines a point p = [a0 : · · · : an] ∈ Pn(k). A variety is basically defined as the set of common zeroes of some polynomials in k[x0, . . . , xn]. For projective varieties, we restrict to homogeneous polynomials, for which vanishing at a point (a0, . . . , an) ∈ kn+1 − {0} is equivalent to vanishing at any point (λa0, . . . , λan) with λ 6= 0, which means that vanishing at a projective point [a0 : · · · : an] is well-defined. Definition 2. The (projective) variety defined by the homogeneous polynomials f1, . . . , fm ∈ k[x0, . . . , xn] is V(f1, . . . , fm) = {[a0 : · · · : an] ∈ Pn(k) | fi(a0, . . . , an) = 0 for 1 6 i 6 m}.