Post-graduate theses
Current Record: 27 of 127
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Identifier |
000451345 |
Title |
Lines on Hypersurfaces |
Alternative Title |
Ευθείες σε υπερεπιφάνειες |
Author
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Λυβιάκης, Εμμανουήλ
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Thesis advisor
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Κουβιδάκης, Αλέξανδρος
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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
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Language |
Greek, English |
Subject |
Algebraic geometry |
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Αλγεβρική γεωμετρία |
Issue date |
2022-07-22 |
Collection
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School/Department--School of Sciences and Engineering--Department of Mathematics and Applied Mathematics--Post-graduate theses
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Type of Work--Post-graduate theses
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Permanent Link |
https://elocus.lib.uoc.gr//dlib/5/d/7/metadata-dlib-1664430850-708310-32055.tkl
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331 |