A k-positive matrix is a matrix where all minors of order k or less are positive. Computing all such minors to test for k-positivity is inefficient, as there are 2n n − 1 of them in an n × n matrix. However, there are minimal k-positivity tests which only require testing n 2 minors. These minimal tests can be related by series of exchanges, and form a family of sub-cluster algebras of the cluster algebra of total positivity tests. We give a description of the sub-cluster algebras that give k-positivity tests, ways to move between them, and an alternative combinatorial description of many of the tests.

We study the geometry of the smooth projective surfaces that are defined by Frobenius forms, a class of homogenous polynomials in prime characteristic recently shown to have minimal possible F-pure threshold among forms of the same degree. We call these surfaces extremal surfaces, and show that their geometry is reminiscent of the geometry of smooth cubic surfaces, especially non-Frobenius split cubic surfaces of characteristic two, which are examples of extremal surfaces. For example, we show that an extremal surface X contains d 2 (d 2 − 3d + 3) lines where d is the degree, which is notable since the number of lines on a complex surface is bounded above by a quadratic function in d. Whenever two of those lines meet, they determine a d-tangent plane to X which consists of a union of d lines meeting in one point; we count the precise number of such "star points" on X, showing that it is quintic in the degree, which recovers the fact that there are exactly 45 Eckardt points on an extremal cubic surface. Finally, we generalize the classical notion of a double six for cubic surfaces to a double 2d on an extremal surface of degree d. We show that, asymptotically in d, smooth extremal surfaces have at least 1 16 d 14 double 2d's. A key element of the proofs is using the large automorphism group of extremal surfaces which we show acts transitively on many sets, such as the set of (triples of skew) lines on the extremal surface. Extremal surfaces are closely related to finite Hermitian geometries, which we recover as the F q 2 -rational points of special extremal surfaces defined by Hermitian forms over F q 2 .

No abstract

Let R be a commutative Noetherian F -finite ring of prime characteristic and let D be a Cartier algebra. We define a self-map on the Frobenius split locus of the pair (R, D) by sending a point P to the splitting prime of (R P , D P ). We prove this map is continuous, containment preserving, and fixes the D-compatible ideals. We show this map can be extended to arbitrary ideals J, where in the Frobenius split case it gives the largest Dcompatible ideal contained in J. Finally, we apply Glassbrenner's criterion to prove that the prime uniformly F -compatible ideals of a Stanley-Reisner rings are the sums of its minimal primes.