There are 27 Lines on a Smooth Cubic Surface Público

Abstract

In the beginning of the nineteenth century, mathematicians became interested in algebraic surfaces in projective spaces. In particular, the remarkable result of Arthur Cayley and George Salmon in $1849$ reveals that there are exactly $27$ lines on every smooth cubic surface in $\PP^{3}_{\mathbb{C}}$. In Chapter $1$, we will work out an example of the Fermat surface. In Chapters $2$ and $3$, we will elaborate on the algebraic geometry machinery, namely fibers, dimensions, Grassmannians, and special classes of morphisms. In Chapter $4$, we will prove the theorem of Cayley and Salmon in the context of modern algebraic geometry, inheriting Salmon's construction of the ``incidence variety'' in his original proof. Meanwhile, we show that there exists a line on every cubic surface in $\PP^{3}_k$.

Table of Contents

1. Introduction ------------------------------------1

1.1 The Fermat Surface----------------------------2

2. Preliminaries------------------------------------6

2.1 Fibers--------------------------- ---------------7

2.2 Dimension -------------------------------------9

2.3 Grassmannians---------------------------------13

3. Morphisms---------------------------------------16

3.1 Projective and finite fibers implies finite---------16

3.2 Flat morphisms----------------------------------19

3.3 Smooth morphisms-------------------------------22

4 Via Dolorosa: The Proof----------------------------25

Bibliography ------------------------------------------34

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Palabra Clave	Algebraic Geometry Scheme Theory Enumerative Geometry
Committee Chair / Thesis Advisor	Parimala Raman, Emory University
Committee Members	Raymond De Luca, Emory University Suresh Venapally, Emory University

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