SECANTS TO THE VARIETY OF COMPLETELY REDUCIBLE FORMS AND THE HILBERT FUNCTION OF THE UNION OF STAR-CONFIGURATIONS

Abstract

We prove that if 𝕏 ≔ 𝕏(t, s) is the union of two linear star-configurations of type t × s for 3 ≤ t ≤ 9 and s ≥ t in ℙ2, then 𝕏 has generic Hilbert function. We also show that (I𝕏)s = {0} when 𝕏 is the union of two linear star-configurations of type s × s for s ≥ 6. Moreover, using the two results with the work in [E. Arrondo and A. Bernardi, On the variety parameterizing completely decomposable polynomials, J. Pure Appl. Algebra215(3) (2011) 201–220], we prove that the secant variety Sec 1( Split s(ℙn)) is not defective for s ≥ 3 and n ≥ 2.