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Identifier

000439278

Title

On the Cayley-Bacharach theorem

Alternative Title

Περί του θεωρήματος Cayley-Bacharach

Author

Φραγκάκη, Αναστασία- Ειρήνη

Thesis advisor

Κουβιδάκης, Αλέξης

Reviewer

Χαραλάμπους, Χαρά
Λουκάκη, Μαρία

Abstract

Cayley-Bacharach theorems are theorems of the following type: describe special sets of points in affine or projective space satisfying the property that hypersurfaces of specific degree which contain a certain subset of those points have to contain the whole set. The history is long and goes back to Pappus' Theorem (320 ac), Pascal's Theorem (1640), Chasles' Theorem (1837) and later on to Cayley's and Bacharach's Theorems, see [10] for a concise overview of the various formulations and generalizations of these type of theorems. In a modern setting, Griffiths and Harris in [9] showed that the above type of theorems can be rephrased in terms of zero loci of sections of a vector bundle. A related question is the following: when does a set of points impose independent conditions on the space of hypersurfaces of specific degree? In other words, given a set of Γ of s points, is the subspace of hypersurfaces of degree k of codimension s in the linear space of hypersurfaces of degree k? This is related to the (s — 1)-ampleness of line bundles, to the Hilbert function of points in the space and has various applications, for example, in the Castelnuovo theory of curves, see [4]. This thesis is based on the paper [3] by D. Eisenbud, M. Green and J. Harris. In the paper the authors review the history of the subject and put the problem in a general algebraic setting, proposing at the end some interesting questions. The aim of the thesis is to focus on the algebraic setting by including all the algebraic background needed and explaining the proofs in detail. For the algebraic background we follow the books (or the class notes) [1], [2], [7], [8].

Language

English

Subject

Algebraic geometry

Αλγεβρική γεωμετρία

Issue date

2021-03-26

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/e/9/f/metadata-dlib-1618475278-907783-13748.tkl