There are 27 Lines on a Smooth Cubic Surface Open Access
Jin, Zihao (Spring 2025)
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
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