William Larsson Krigholm: On the Annihilator of the Complete Symmetric Form

Abstract: In this thesis, we study a conjecture concerning the annihilator ideal of a complete symmetric form of odd degree. More precisely, let \(H_{2d+1}\) be the complete symmetric form of degree \(2d+1\), and consider its annihilator under the apolar action. The conjecture predicts a parity-dependent pattern for the minimal generators of this ideal: when d is odd, no additional minimal generator should appear in degree \(d+2\), while when d is even, exactly one such generator should appear.

We prove the conjecture in codimension 3, using a parity argument together with a theorem of Buchsbaum–Eisenbud on grade three Gorenstein ideals. We also prove the conjecture for all codimensions when d is odd. Finally, we establish a general reduction result showing that, in the remaining even case, the possible degree \(d+2\) generators span a vector space of dimension at most one.