On the K-theory of planar cuspical curves and a new family of polytopes

Abstract

Let k be a regular F_p-algebra, let A = k[x,y]/(x^b - y^a) be the coordinate ring of a planar cuspical curve, and let I = (x,y) be the ideal that defines the cusp point. We give a formula for the relative K-groups K_q(A,I) in terms of the groups of de Rham-Witt forms of the ring k. At present, the validity of the formula depends on a conjecture that concerns the combinatorial structure of a new family of polytopes that we call stunted regular cyclic polytopes. The polytopes in question appear as the intersections of regular cyclic polytopes with (certain) linear subspaces. We verify low-dimensional cases of the conjecture. This leads to unconditional new results on K_2 and K_3 which extend earlier results by Krusemeyer for K_0 and K_1.