Introduction to enumerative geometry.

Lecture Register

Academic year 2023/2024
Lecturer Angelo Vistoli

Lecture

  • 07 Mar 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Introduzione al corso. Esempi di problemi enumerativi. Cicli su varietà, e gruppi di Chow. La successione di omotopia.

  • 08 Mar 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Funtorialità dei gruppi di Chow: pushforward proprio e pullback piatto.

  • 14 Mar 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Compatibilità tra pushforward proprio e pullback piatto. Esempi.

  • 15 Mar 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Anelli di Chow di varietà lisce. Anelli di Chow di spazi proiettivi. Formula di proiezione. Esempi. Il grado di una varietà di Veronese.

  • 04 Apr 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Online

    Anelli di Chow di prodotti. Grado dell'immersione di Segre. Soluzioni di problemi enumerativi per curve piane riducibili.

  • 05 Apr 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Grado della varietà discriminante. Fibrati vettoriali e loro proiettivizzazioni.

  • 11 Apr 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Chern classes and their properties. Examples and applications.

  • 12 Apr 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Proof of the splitting principle, and of the multiplicativity of Chern classes. Grassmanianns: definitions, and first properties.

  • 18 Apr 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Schubert cycles, and Schubert cells. The Chow ring of G(2,4).

  • 26 Apr 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    The Chow ring of the Grassmannian G(2,4).

  • 02 May 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    27 lines on a cubic surface.

  • 03 May 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Chow rings of general Grassmannians. Intersection of Schubert cycles in complementary dimension. Pieri's rule. Giambelli's formula. Examples.

  • 09 May 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Chern classes of the tautological and the quotient bundles on Grassmannians. Linear subspaces on quadrics.

  • 10 May 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Preparazione ore lezioni - Products of Schubert cycles on G(2,n). Degree of the Plücker embedding of G(2,n). Fano varieties of lines; general setup.

  • 16 May 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    The universal Fano variety. The class of the universal Fano variety in the product of the Grassmannian with projective space. 2875 lines on a quintic threefold. The degree of the variety of quartic surfaces in P^3 containing a line is 320.

  • 17 May 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    The number of conics on a general quintic threefold. Hilbert schemes: definition and first properties. The Grassmannian as a Hilbert scheme.

  • 23 May 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Hilbert schemes of hypersurfaces. Tangent spaces to a scheme.

  • 24 May 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Linear functors and functors preserving finite products. Tangent spaces to Grassmannians and Hilbert schemes.

  • 30 May 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    Properties of normal bundles. Fano schemes of lines. Tangent spaces to Fano schemes, and applications.

  • 31 May 2024 (2h 00m)

    Angelo Vistoli - Course (teaching activity) - Both face to face and online

    The main theorem on tangent spaces to Fano varieties of lines on hypersurfaces.