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Algebraic Geometry

Moduli Spaces of Curves

  • Deformation theory of complex manifolds, basic notions.
  • Teichmüller spaces T_g, mapping class groups Map_g and the construction of M_g as quotient T_g/Map_g.
  • The theorem of Teichmüller.
  • Stable homology of M_g.

Intersection Theory

  • Algebraic cycles, flat pullback, proper pushforward.
  • Rational and algebraic equivalence. Chow groups.
  • Degree of a zero cycle. Numerical equivalence.
  • Vector bundles, cell decompositions.
  • Pseudodivisors, first Chern class of a line bundle.
  • Chern and Segre classes.
  • Cones, abelian cones, normal cones.
  • Degeneration to the normal cone.
  • Gysin pullback.
  • Properties of the Gysin pullback.
  • Statement of Grothendieck-Riemann-Roch and applications. If time allows,sketch of proof.

Algebraic Geometry

Il corso coprirà il materiale descritto nelle sezioni 1-8 del secondo capitolo del testo di R. Hartshorne, Algebraic Geometry, GTM 52. In dettaglio: fasci, schemi, sottoschemi, proprietà degli schemi e dei loro morfismi, criteri valutativi, fasci coerenti e quasicoerenti, fibrati, fascio cotangente relativo e assoluto, fibrati lineari e divisori, morfismi proiettivi e loro proprietà.

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