Research Group:

## 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.