Richard P. Stanley Seminar in Combinatorics: Counting curves on P^r, degenerations of torus orbits, and beyond

We will discuss the following problem in enumerative geometry: if C is an algebraic curve, p_1,…,p_n are points on C, and X_1,…,X_n are linear subspaces of the projective space P^r, then how many maps f:C\to P^r are there with the property that f(p_i)\in X_i? It turns out that the problem can be reduced to the calculation of cohomology classes of certain subvarieties of Grassmannians. In the special case where the X_i are all points, these subvarieties are torus orbit closures, which are well-understood. The general case is more mysterious. We will describe degeneration techniques which give some new insight into the geometry of orbit closures, and also lead to a complete answer to the general problem when r=2 in terms of SSYT avoiding certain patterns. Similar techniques work in principle in more generality, but there are combinatorial obstacles.

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For more info, see https://math.mit.edu/combin/