Lines on Cubic Surface Research Report in Mathematics, Number 04, 2021

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 de�ned by the projection (X, �) �→ X where � 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 B�6 : B�6P2 ��� X.