Grupo de Galois/Monodromia: cálculo dos pontos de inflexão de cúbicas planas e retas em superfícies cúbicas não singulares

Abstract

In 1851 the french mathematician Charles Hermite showed that the Galois group and monodromy coincide in the context of algebraic geometry. However, it was Joe Harris (in 1979) that presented this result and applications in a modern language. In this work initially we introduce the Galois group and monodromy in the context of algebraic geometry following the lines of the article "Galois groups of enumerative problems"[10]. Next we determine the monodromy group of the inflection points of a non-singular plane cubic. Since this group is soluble, we determine the coordinates of these points from the Weierstrass equation of such cubic. We conclude by reviewing Harris'results about the monodromy group of lines in non-singular cubic surfaces, emphasizing the calculation of these lines from the equation that defines a non-singular cubic surface containing three lines two by two prefixed disjoint (according Mckean-Minaham-Zhang, 2020 ([16])).