SOME APPLICATIONS OF THE UNION OF STAR-CONFIGURATIONS IN Pn

Abstract

It has been proved that if X(s,s) is the union of two linear star-configurations in P2 of type s × s, then (IX(s,s) )s (cid:54)= {0} for s = 3, 4, 5, and (IX(s,s) )s = {0} for s ≥ 6. We extend P2 to Pn and show that if X(s,s) is the union of two linear star-configurations in Pn, then (IX(s,s) )s = {0} for n ≥ 3 and s ≥ 3. Using this generalization, we also prove that the secant variety Sec1(Splits(Pn)) has the expected dimension 2ns + 1 for n ≥ 3 and s ≥ 3.

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