2009/11/25 by Knutson, Allen · 1 citation
#13P10 #14D06 #14G05 #14M15 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.0911.4941
Let f be a polynomial of degree n in ZZ[x1,..,xn], 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 \FFpn is not a multiple of p, then all these intersections in Ån\FFp just described are reduced. (If this holds for infinitely many p, then it holds over \QQ as well.) More specifically, there is aFrobenius splitting_ on Ån\FFp compatibly splitting all these subschemes Y. We determine when a Gröbner degeneration f0=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 ∏i=1n xi. Then for all large p, there is a Frobenius splitting that compatibly splits f's hypersurface and all the associated Y. The Gröbner 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 Gröbner basis. We exhibit an f whose associated Y include Fulton's matrix Schubert varieties, and recover much more easily the Gröbner basis theorem of [Knutson-Miller '05]. We show that in Bott-Samelson coordinates on an opposite Bruhat cell Xv_∘ in G/B, the f defining the complement of the big cell also has initial term ∏i=1n xi, and hence the Kazhdan-Lusztig subvarieties Xvw∘ degenerate to Stanley-Reisner schemes. This recovers, in a weak form, the main result of [Knutson '08].