Torus orbits on homogeneous varieties and Kac polynomials of quivers

In this paper we prove that the counting polynomials of certain torus orbits in products of partial flag varieties coincides with the Kac polynomials of supernova quivers, which arise in the study of the moduli spaces of certain irregular meromorphic connections on trivial bundles over the projective line. We also prove that these polynomials can be expressed as a specialization of Tutte polynomials of certain graphs providing a combinatorial proof of the non-negativity of their coefficients

On Hyperfocused Arcs in PG(2,q)

A k-arc in a Dearguesian projective plane whose secants meet some external line in k-1 points is said to be hyperfocused. Hyperfocused arcs are investigated in connection with a secret sharing scheme based on geometry due to Simmons. In this paper it is shown that point orbits under suitable groups of elations are hyperfocused arcs with the significant property of being contained neither in a hyperoval, nor in a proper subplane. Also, the concept of generalized hyperfocused arc, i.e. an arc whose secants admit a blocking set of minimum size, is introduced: a construction method is provided, together with the classification for size up to 10