A Simplified Game for Resolution of Singularities

We will describe a combinatorial game that models the problem of resolution of singularities of algebraic varieties over a field of characteristic zero. By giving a winning strategy for this game, we give another proof of the existence of resolution

Counting projections of rational curves

Given two general rational curves of the same degree in two projective spaces, one can ask whether there exists a third rational curve of the same degree that projects to both of them. We show that, under suitable assumptions on the degree of the curves and the dimensions of the two given ambient projective spaces, the number of curves and projections fulfilling the requirements is finite. Using standard techniques in intersection theory and the Bott residue formula, we compute this number.Comment: 27 page