We give a new and simple proof that unmixed local rings having Hilbert-Kunz multiplicity equal to 1 must be regular.

We characterize Beauville surfaces of unmixed type with group either PSL(2, p) or PGL(2, p), thus extending previous results of Bauer, Catanese and Grunewald, Fuertes and Jones, and Penegini and the author.

A smooth algebraic surface S is said to be isogenous to a product of unmixed type if there exist two smooth curves C, F and a finite group G, acting faithfully on both C and F and freely on their product, so that S = (C × F )/G. In this paper we classify the surfaces of general type with pg = q = 1 which are isogenous to an unmixed product, assuming that the group G is abelian. It turns out tha...

If C is a clutter with n vertices and q edges whose clutter matrix has column vectors A = {v1, . . . , vq}, we call C an Ehrhart clutter if {(v1, 1), . . . , (vq , 1)} ⊂ {0, 1} n+1 is a Hilbert basis. Letting A(P ) be the Ehrhart ring of P = conv(A), we are able to show that if A is the clutter matrix of a uniform, unmixed MFMC clutter C, then C is an Ehrhart clutter and in this case we provide...

Consider a regular triangulation of the convex-hull P of a set A of n points in Rd, and a real matrix C of size d×n. A version of Viro’s method allows to construct from these data an unmixed polynomial system with support A and coefficient matrix C whose number of positive solutions is bounded from below by the number of d-simplices which are positively decorated by C. We show that all the d-si...