“When [Oscar Zariski] spoke the words algebraic variety, there was a certain resonance in his
voice that said distinctly that he was looking into a secret garden. I immediately wanted to be
able to do this too ... Especially, I became obsessed with a kind of passion flower in this garden,
the moduli spaces of Riemann.” David Mumford
Curvas algebraicas son un objeto de estudio clásico en geometría algebraica que todavía
atrae mucha investigación e interés, en particular la geometría de sus espacios de moduli.
Mi intención es hacer una introducción básica sobre curvas algebraicas y sus espacios de
moduli enfocada en ejemplos, preguntas y ojalá discusion abierta. Estudiantes terminando
el pregrado o en etapa temprana del posgrado están particularmente invitados.
El curso consiste en 5 sesiones de 1 hora.
Lunes 29: Weil/Cartier divisors on curves, linear series, line bundles, morphisms to
projective spaces, Riemann-Roch, and Geometric Riemann-Roch.
Martes 30: Ramification, Riemann-Hurwitz and monodromy. Positivity of line bundles
on curves. Generic curves of low genus. Examples.
Miércoles 1: First approach to Mg , dimension, rationality/unirationality/uniruledness,
and Severi’s theorem.
Jueves 2: Stable curves, divisors on Mg , and the Brill-Noether locus.
Viernes 3: Grothendieck-Riemann-Roch, the canonical class of Mg , and Harris-Mumford
Theorem.
Referencias sugeridas:
[AC] E. Arbarello and M. Cornalba, Footnotes to a paper of Beniamino Segre, Mathematische Annalen 256
(1981), 341–362.
[ACGH] E. Arbarello, M. Cornalba, P. Griffiths and J. Harris, Geometry of algebraic curves, Grundlehren der
mathematischen Wissenschaften 267, Springer.
[F] G. Farkas, Aspects of the birational geometry ofMg , in: Geometry of Riemann surfaces and their moduli
spaces, Surveys in Differential Geometry Vol. 14 (2010), 57–111.
[HM] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Inventiones Math. 67
(1982), 23–88.
[Ha] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics. Springer-Verlag, New York 1977.