We compute the linear strand of the minimal free resolution of the ideal generated by k x k sub-permanents of an n x n generic matrix and of the ideal generated by square-free monomials of degree k. The latter calculation gives the full minimal free resolution by work of Biagioli-Faridi-Rosas. Our motivation is to lay groundwork for the use of commutative algebra in algebraic complexity theory. We also compute several Hilbert functions relevant for complexity theory.
Let A be a union of smooth plane curves C_i, such that each singular point of A is quasihomogeneous. We prove that if C is a smooth curve such that each singular point of A U C is also quasihomogeneous, then there is an elementary modification of rank two bundles, which relates the O_{P^2} module Der(log A) of vector fields on P^2 tangent to A to the module Der(log A U C). This yields an inductive tool for studying the splitting of the bundles Der(log A) and Der(log A U C), depending on the geometry of the divisor A|_C on C.
Alfeld introduced a subdivision AS(n) of an n-simplex, generalizing the Clough-Tocher split of a triangle. A formula for the dimension of the spline space C^r_k(AS(n)) was conjectured recently by Foucart-Sorokina. We prove that the graded module of C^r-splines on the cone over AS(n) is isomorphic to the module D^{r+1}(A_n) of multiderivations on the type A_n Coxeter arrangement. A theorem of Terao shows that the module of multiderivations of a Coxeter arrangement is free and gives an explicit basis. As a consequence the conjectured formula holds.
Let U be a basepoint free four-dimensional subpace of the space of sections of bidegree (a,b) on X = P^1 x P^1, with a and b at least 2. The sections corresponding to U determine a regular map from X to P^3. We show that there can be at most one linear syzygy on the associated bigraded ideal I_U in the bigraded ring k[s,t;u,v]. Existence of a linear syzygy, coupled with the assumption that U is basepoint free, implies the existence of an additional special pair of minimal first syzygies. Using results of Botbol, we show that these three syzygies are sufficient to determine the implicit equation of the image of X in P^3; we also show that the singular locus must contain a line.
