La geometria delle funzioni di Hilbert: il caso zero-dimensionale

In the classification of Algebraic Varieties it is useful to take into account both ¿discrete¿ and ¿continuous¿ invariants. For example, the isomorphism class of a nonsingular projective curve C can be identified by giving the genus g and a point on the moduli space of genus g curves. The Hilbert function of a variety X is an example of a discrete in- variant: it is a sequence of non negative integers, which eventually coincides with a polynomial with rational coefficients, the Hilbert polynomial of X. The coefficients of the Hilbert polynomial encode important information on X, such as its dimension and arithmetic genus. Due to the algebraic nature of its definition, the Hilbert function can be defined also for subschemes of projective space Pn. One can ask what is the geometric meaning of the integers appear- ing in the Hilbert function before it coincides with the Hilbert polynomial. It turns out that a particular behavior in the Hilbert function of a scheme X can force the existence of distinguished subschemes of X. This thesis focuses on the case of zero-dimensional reduced schemes, and how the behavior of the Hilbert function allows one to detect a large subscheme for example on a hypersurface, a linear space or a curve. The first chapter is an introduction to the Hilbert functions with particular emphasis on their basic properties in the case of finite reduced sets of points. A characterization of such Hilbert function, due to P. Maroscia, is stated and proved at the end of the chapter. The second chapter, following [BGM94], explores the consequence of maximal growth (see Theorem 1.11) of the Hilbert function of a zeroscheme X. In some cases it is possible to determine an ideal J defining a subscheme Y of X, and also calculate its Hilbert function, which can be exploited to yield further geometrical information on Y (see The- orem 2.15 or Theorem 2.16). The hypotheses in Theorem 2.16 can be weakened, and to do so it is necessary to consider the generic initial ideal (gin) with respect to the revlex order: it is a monomial ideal canonically associated to a closed subscheme of Pn (indeed, to any homogeneous ideal of the polynomial ring k[x0, . . . , xn]). Despite being substantially a combinatorial object, the gin of an ideal defining a scheme X carries significant geometrical information on X. This is the content of the third chapter, the main reference being [AM07]. The last chapter deals with particular schemes in P2, introduced in [RZ09]. Since Hilbert's theorem on syzygies, it is well known that homogeneous ideals have a finite graded free resolution: in the case of the projective plane, zero-dimensional schemes have defining ideals whose resolution can be fully described. Then a particular class of schemes is considered, namely those which resolution has a particular shape. An appendix on some purely algebraic results quoted in the other chapters, such as a proof of Hilbert's syzygy Theorem, is included at the end.