## Tesi etd-05222018-151008 |

Thesis type

Tesi di laurea magistrale

Author

FAGIOLI, FILIPPO

URN

etd-05222018-151008

Title

Counting Lines on Projective Hypersurfaces via Characteristic Classes

Struttura

MATEMATICA

Corso di studi

MATEMATICA

Commissione

**relatore**Prof.ssa Pardini, Rita

Parole chiave

- orientation
- Chern classes
- vector bundles
- Euler class
- lines
- hypersurfaces
- characteristic classes
- Enumerative Geometry

Data inizio appello

08/06/2018;

Consultabilità

secretata d'ufficio

Riassunto analitico

In this thesis we discuss the enumerative geometry problem of counting lines on projective hypersurfaces via characteristic classes.

In the first part, divided into two chapters, we recall some necessary prerequisites on real and complex vector bundles, and prove key properties of Euler and Chern classes in order to solve the enumerative problem.

In the second part, also divided into two chapters, we approach the enumerative problem both in the complex and in the real case by applying the tools presented in the first part. Our investigation leads to the following conclusions. In the complex case, the top Chern class of an appropriate bundle provides the exact number of lines contained in a general hypersurface of degree 2n-3 in the n-dimensional projective space. In the real case, the same methodology is also applied by making use of the Euler class. This leads to a non trivial lower bound for the total number of lines on a general hypersurface of degree 2n-3 in the real n-dimensional projective space.

In the first part, divided into two chapters, we recall some necessary prerequisites on real and complex vector bundles, and prove key properties of Euler and Chern classes in order to solve the enumerative problem.

In the second part, also divided into two chapters, we approach the enumerative problem both in the complex and in the real case by applying the tools presented in the first part. Our investigation leads to the following conclusions. In the complex case, the top Chern class of an appropriate bundle provides the exact number of lines contained in a general hypersurface of degree 2n-3 in the n-dimensional projective space. In the real case, the same methodology is also applied by making use of the Euler class. This leads to a non trivial lower bound for the total number of lines on a general hypersurface of degree 2n-3 in the real n-dimensional projective space.

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