Frobenius splitting, point-counting, and degeneration

Abstract

Let f be a polynomial of degree n in ZZ[x_1,..,x_n], typically reducible but squarefree. From the hypersurface {f=0} one may construct a number of other subschemes {Y} by extracting prime components, taking intersections, taking unions, and iterating this procedure. We prove that if the number of solutions to f=0 in \FF_p^n is not a multiple of p, then all these intersections in \AA^n_{\FF_p} just described are reduced. (If this holds for infinitely many p, then it holds over \QQ as well.) More specifically, there is a_Frobenius splitting_ on \AA^n_{\FF_p} compatibly splitting all these subschemes {Y}. We determine when a Gr\"obner degeneration f_0=0 of such a hypersurface f=0 is again such a hypersurface. Under this condition, we prove that compatibly split subschemes degenerate to compatibly split subschemes, and stay reduced. Our results are strongest in the case that f's lexicographically first term is \prod_{i=1}^n x_i. Then for all large p, there is a Frobenius splitting that compatibly splits f's hypersurface and all the associated {Y}. The Gr\"obner degeneration Y' of each such Y is a reduced union of coordinate spaces (a Stanley-Reisner scheme), and we give a result to help compute its Gr\"obner basis. We exhibit an f whose associated {Y} include Fulton's matrix Schubert varieties, and recover much more easily the Gr\"obner basis theorem of [Knutson-Miller '05]. We show that in Bott-Samelson coordinates on an opposite Bruhat cell X^v_\circ in G/B, the f defining the complement of the big cell also has initial term \prod_{i=1}^n x_i, and hence the Kazhdan-Lusztig subvarieties {X^v_{w\circ}} degenerate to Stanley-Reisner schemes. This recovers, in a weak form, the main result of [Knutson '08].