Classificação de retas e planos no espaço projetivo das cônicas e algumas aplicações

Abstract

According to the article Nets of conics and associated Artinian algebras of length 7 ([1]) by Abdallah, Emsalem, and Iarrobino, this work classifies the orbits under the action of the automorphism group of the complex projective plane, Aut(P 2 ), on the Grassmannians G2(S2) and G3(S2), where S2 denotes the space of homogeneous polynomials of degree 2 in three variables with complex coefficients. First, we present preliminary notions on group actions in projective spaces and Grassmannians. Next, we develop the complete classification of orbits in G2(S2), determining their dimensions and specialization relations. The analysis extends to G3(S2) (whose elements are called nets of conics), highlighting the geometric identification of planes associated with conic nets and their intersections with the hypersurface H of singular conics. Finally, we discuss applications (of the classification obtained for G3(S2)) to classifying degree-4 morphisms in the projective plane and classifying local graded Artinian algebras with Hilbert function (1, 3, 3, 0).