Pontos fechados no esquema de Hilbert de curvas Hilb at+bP³ com (a,b) em {(1,1), (1,2), (2,1), (2,2)}

Abstract

The main objective of this work is to make an explicit description of the closed points in the Hilbert schemes of Hilbat+bP 3 Curves with (a, b) ∈ {(1, i),(2, i)} 2 i=1. For this, we make a brief theoretical review about modules, graduated modules and related concepts, to demonstrate the Hilbert-Serre Theorem, obtaining as a corollary the existence of the Hilbert polynomial. Next, we touch on schema theory, moving on to the concept of Gotzmann representation (and Gotzmann number), which we use to indicate how to visualize Hilbert schemata, as closed in certain grassmannians (cf. Proposition 2.9). In the final part of the text, we include some results involving degree of varieties and the Hilbert polynomial calculus for certain schematic unions, which we use to achieve our main objective.