Aut(P^3) - Órbitas do esquema de Hilbert, Hilb^{at+b}P^3 com a, b em {1, 2}

Abstract

The Hilbert scheme, introduced by Alexander Grothendieck in the 1960s, is a funda- mental object in algebraic geometry whose closed points parametrize closed subschemes of a projective scheme with a fixed Hilbert polynomial. This dissertation investigates the stratification of the Hilbert scheme of curves in P3, denoted by Hilb at+b P3 with a, b ∈ {1, 2}, by analyzing the orbits generated under the action of the automorphism group Aut(P3). The study focuses on the Hilbert polynomials: t + 1, t + 2, 2t + 1, and 2t + 2, which correspond geometrically to lines, the schematic union of a line and a point, conics, and the schematic union of disjoint lines or of a conic with a point in P3. Using techniques from commutative algebra, geometric invariant theory, and com- putational support from the software Macaulay2, we present an explicit classification of these orbits, the computation of their dimensions, and the description of specializa- tion relations (degenerations) among them. The results demonstrate that these spaces decompose into a finite number of orbits, revealing a well-defined geometric hierar- chy that reflects the relative position of irreducible components and the non-reduced structures of the studied curves.