Junzo Watanabe: The resultants of quadratic complete intersections and the higher Hessians of n-ary n-ic

Abstract

If n linearly independent homogenous forms are given in the polynomial ring in n variables, we know that it is a complete intersection by the non-vanishing of the resultant of these forms. The SLP of an Artinian Gorenstein algebra A can be determined by the non-vanishing of the Higher Hessians of the Macaulay dual generator F of A. If A is a quadratic complete intersection, F is a polynomial in n variables of degree n. On the other hand if F is a homogeneous form of degree n in n variables, the annihilator of F contains n quadratic forms. This gives us a criterion for the Ann(F) to be a complete intersection. Keeping these criteria in mind, I will make an attempt to construct a counter example to the long standing conjecture “If A is a ci, then A should have the SLP”.