Abstract
In this dissertation, we enumerate the 27 lines on a smooth cubic surface X ⸦ P3. We do this by understanding the combinatorics of the subset S of disjoint lines on X of the Grassmanian Gr (2,4) of lines on P3. Further, using the incidence correspondence defined by the projection (X, l) �→X where l is a line on X, we show that the relation is a 27- sheeted covering map by studying the inverse image of lines on a smooth cubic X under the blowup map
Table of Contents
Abstract ii
Declaration and Approval iv
Dedication vii
Acknowledgments ix
Chapter 1: Introduction
Chapter 2: Preliminaries
2.1 What is Algebraic Geometry? 3
2.1.1 Affine and Projective Algebraic Varieties 3
2.1.2 The Zariski Topology on Affine Varieties 5
2.1.3 The IdealI-VarietyVCorrespondence 7
2.1.4 Nonsingularity of Algebraic Varieties 9
2.2 Blowup of Algbraic Varieties at Points 11
Chapter 3: The Geometry of Lines on Smooth Cubics inP 3
3.1 The GrassmannianGr(2,4)of lines inP 3 and Some Useful Enumerative Lemmas 14
3.1.1 The Plűcker embedding 15
3.1.2 Irreducibility 17
3.2 Smooth Cubic is birational toP 1 ×P 1 or toP 2 21
3.3 Blowup ofP 2 at 6 Points in General Position 21
Chapter 4: The 17+10 or the 6+15+6 Lines on Smooth Cubic 23
4.1 17+10 Lines on a Smooth Cubic inP 3 29
4.2 6+15+6 Lines on a Smooth Cubic inP 3 32
Bibliography 34
Chapter 1
Introduction
The main goal of this thesis is to study the configuration of lines on a nonsingular complex cubic surfaces and be able to demonstrate that there are27lines on such a surface.
We do this in two ways
(i) We explore the geometry of Grasmaniann Gr (2,4) of Lines on P3 through the Plűcker embedding
Gr (2,4)�→P5
and the representation of lines in Gr (2,4). We then show that there is at least a line on a cubic surface X in P3 and that one can find two such lines l and m which are disjoint on X. Further, we demonstrate that the set of lines intersecting a given arbitrary line l in P3 is a subset of Gr (2,4) and that the set has exactly5pairs of disjoint lines. We also appreciate that if one line of the5pairs of lines that intersect l also intersects a line m disjoint to l, then 5 disjoint lines of the five pairs of lines also do intersect m.
This gives 17 disjoint lines on X:l,m, the 5 disjoint lines intersecting both l and m, the 5 disjoint lines that only intersect l and the other 5 disjoint lines that only intersect m. We then convince the reader and ourselves that any line on X\ {The 17 lines above} would intersect exactly 3 of the five lines intersecting the line l; for if more, the line would be l or m and if less then the line would intersect at least 3 of the five lines. There are therefore �5� = 10 disjoint lines on X\ {The 17 lines above}.
(ii) we appreciating that a smooth cubic is birational to P2 and that the blow up Bl6P2 of P2 at 6 points in general position is isomorphic
Bl6 :Bl6P2 ���X
to a smooth cubic X. We further demonstrate that the space of smooth cubics is a dense open subset U of P19. Now by taking a subset M of the product U×Gr (2,4) consisting of pairs (X,l) of a smooth cubic X and a Line l on X, we define the incidence correspondence
π:M→U
by the projection (X,l)�→X. The goal of the thesis is then equivalent to counting the number of inverse images ofπ; that is, showing thatπis a 27-sheeted covering map. Finally, we count these inverse images ofπby counting the images under the blow-up mapBl6,of the 6 exceptional hypersurfaces, the strict transform of 6 := 15 lines through any two of the six blown up points and the strict transform of �6� := 6 conics through any five of the six blown up points.
The outline of the thesis is as follows:
Chapter 2:
In this chapter, we introduce algebra-geometry dictionary and the geometric object of our study: Irreducible Smooth cubic.
Chapter 3:
This chapter sets up the stage for the thesis’ enumerative goal by focusing on notions such as blowup of P2 at 6 points, space of smooth cubics and the geometry of Grassmanian Gr (2,4) of lines on P3.
Chapter 4:
Here, we enumerate lines on smooth cubic in two ways: though combinatorics of lines on smooth cubic or through the analysis of the preimage, through the blowup map
Bl6 :Bl6P2 ���X,
of lines on the smooth cubic X.
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