The Fano surface S of a smooth cubic threefold F into P4 carries many remarkable geometric properties.
By example, one can recover the fact that the cotangent space of S is generated by global sections.
The threefold F' is isomorphic to F. Thus knowing a Fano surface S, we can recover the threefold F. By the Tangent Bundle Theorem, we can also understand geometrically the invariants of S: a) Recall that the second Chern number of a rank 2 vector bundle on a surface is the number of zeroes of a generic section.
For a Fano surface S, a 1-form w defines also a hyperplane section {w=0} into P4 of the cubic F. The zeros of the generic w on S corresponds bijectively to the numbers of lines into the smooth cubic surface intersection of {w=0} and F, therefore we recover that the second Chern class of S equals 27. b) Let w1, w2 be two 1-forms on S. The canonical divisor K on S associated to the canonical form w1 ∧ w2 parametrizes the lines on F that cut the plane P={w1=w2=0} into P4.
Consider the linear span of Ls and Lt : it is an hyperplane into P4 that cuts F into a smooth cubic surface.