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Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Juan Carlos Naranjo del Val, [en] In Algebraic Geometry numbers 27 and 28 are usually associated with two well-known classical results. All smooth cubic surfaces contain 27 distinct lines. And all smooth plane quartics have 28 bitangents. The aim of this work is to stablish a relation between these two statements. First, we have introduced the theoretical basis needed to demonstrate the two classical results. In the final part, we have suggested a method with which the 27 lines contained in a cubic surface can be transformed into bitangents of a plane quartic and, also from the surface, an additional bitangent can be formed, so that we ultimately obtain the 28 bitangents.

Treballs Finals de Grau de Matemàtiques de la Facultat de Matemàtiques de la Universitat de Barcelona, Any: 2012 , Director: Juan Carlos Naranjo del Val, This work, which consists in two separate parts, will attempt to build an equation for a non-singular plane projective quartic over an algebraically closed eld, namely C. In the course of this trail, algebric and geometric theory will be introduced and discussed here, taking special care in subjects such as Bézout's theorem, the divisor language, Riemann-Roch's theorem and some matters about symplectic algebra. This path will lead us to Steiner-Hesse's theorem which provides a way to write an equation for a non-singular quartic. Finally, we will use those methods in order to attempt the development of an equation for the Klein quartic. An appendix lies at the end of the work talking about a pair of questions which can be investigated parting from the subjects in the main text, and a brief historical background for the things shown on it.

Let $C$ be a smooth integral projective curve admitting two pencils $g^1_a$ and $g^1_b$ such that $g^1_a + g^1_b$ is birational. We give conditions in order that the complete linear system $\vert sg^1_a + rg^1_b\vert$ be normally generated or very ample.

Sea X una curva algebraica irreducible sobre un cuerpo algebrai­camente cerrado k y sea X' un modelo no singular de X. Designemos por 0 (0') el haz de anillos locales de X (X'), 0x (0.,') será su fibra en el punto x y k (X) = k (X') el cuerpo de funciones racionales de ambas curvas. Llamemos g al género (efectivo) de X, es decir, g es el género de la extensión k -+ k (X) en el sentido de CHEV ALLEY: 1 g = dimk H (X', X 0')...