Brill-noether Theory, Ii

This article follows the paper of Griffiths and Harris, "On the variety of special linear systems on a general algebraic curve." 1. WARMUP ON DEGENERATIONS The classic first problem in Schubert calculus is: how many lines intersect four general lines in P 3 ? First, what does this numerology come from? The space of lines in P 3 is G(1, 3) which has dimension 2 × 2 = 4. The locus of lines intersecting a given one in P 3 is a hypersurface in G(1, 3)(we'll expand more on this later, but there are various easy ways to see it in this case), or in other words the condition that a line intersect a given line is of codimension 1. Therefore, the locus of lines intersecting four general lines in P 3 should be of dimension 0, so one can ask for its degree. One quick and dirty way to solve such problems is via degeneration. Note that by Bertini's theorem, the intersection problem is guaranteed to be transverse for general choices of lines. The idea is that as the fixed line(s) vary in G(1, 3), the answer should vary continuously as long as it is well-defined, i.e. as long as the intersection is transverse. But then we may gain some leverage by picking special configurations of lines. For instance, suppose that we degenerate two of the lines, say 1 and 2 , so that they intersect at a point p 12. Then any line meeting both 1 and 2 must either meet them at the same point p 12 , or at separate points, in which case it lies in the plane Λ 12 spanned by them. Similarly, suppose we degenerate the other two lines 3 and 4 to intersect at a point p 34. Then any line meeting both 3 and 4 must pass through p 34 or lie in the plane Λ 34 spanned by them (and these conditions are also sufficient). Now how many lines meet all four of 1 , 2 , 3 , 4 ? There are four cases to check. If the line passes through p 12 and p 34 , then it is the unique line spanned by them. If it lies in Λ 12 and Λ 34 , then it is the unique line which is their intersection. It cannot lie in Λ 12 and pass through p 34 as long …