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Identifier

000451345

Title

Lines on Hypersurfaces

Alternative Title

Ευθείες σε υπερεπιφάνειες

Author

Λυβιάκης, Εμμανουήλ

Thesis advisor

Κουβιδάκης, Αλέξανδρος

Abstract

In this thesis we study lines contained in hypersurfaces of the complex projective space. These are the vanishing loci of homogeneous polynomials in n + 1 variables. The degree of the hypersurface is the degree of the polynomial. We expect that the more lines the hypersurface contains, the more close to a hyperplane should be. In fact, the lines contained in a hyperplane of P n are parametrized by the Grassmannian G(1, n−1), a variety of dimension 2(n−2). On the other hand, one can show that the locus of lines contained in an irreducible hypersurface of P n has dimension ≤ 2(n − 2) and the equality holds exactly when the hypersurface is a hyperplane. The above can be rephrased as follows. A hyperplane is an irreducible hypersurface of degree one. So we should expect, as the degree increases the dimension of the locus of lines contained in the hypersurface to decrease and at some point this locus to become empty. This is not quite true. For degree one, we have just one hypersurface, the hyperplane. But for higher degrees this is not the case, and we may find special hypersurfaces containning unexpectedly many lines. But for what we call the general hypersurface of specific degree, we shall see that this is true. Throughout this thesis we work over the complex numbers. We denote by P n the complex projective space of dimension n. We assume some basic familiarity with Algebraic Geometry: projective space, projective varieties, line bundles and vector bundles on manifolds etc. With the term variety we always mean a projective variety, except if otherwise stated. In the first three chapters we recall in short some basic theory: the Chow ring of a smooth variety, the Grassmannian and the theory of Chern classes. In chapter four we study lines on hypersurfaces and in the last chapter we specialize to the case of cubics (hypersurfaces of degree 3), which is a central subject of study on its own. The main references we use are [2] and [4

Language

Greek, English

Subject

Algebraic geometry

Αλγεβρική γεωμετρία

Issue date

2022-07-22

Collection

School/Department--School of Sciences and Engineering--Department of Mathematics and Applied Mathematics--Post-graduate theses

Type of Work--Post-graduate theses

Permanent Link

https://elocus.lib.uoc.gr//dlib/5/d/7/metadata-dlib-1664430850-708310-32055.tkl