Gorenstein algebras of embedding dimension four : Components of P Gor ( H ) for H = ( 1 , 4 , 7 , . . . , 1 )

A Gorenstein sequence H is a sequence of nonnegative integers H = (1, h1, . . . , hj = 1) symmetric about j/2 that occurs as the Hilbert function in degrees less or equal j of a standard graded Artinian Gorenstein algebra A = R/I , where R is a polynomial ring in r variables and I is a graded ideal. The scheme PGor(H) parametrizes all such Gorenstein algebra quotients of R having Hilbert function H and it is known to be smooth when the embedding dimension satisfies h1 ≤ 3. The authors give a structure theorem for such Gorenstein algebras of Hilbert function H = (1, 4, 7, . . .) when R = K[w, x, y, z] and I2 ∼= 〈wx, wy, wz〉 (Theorem 3.7, 3.9). They also show that any Gorenstein sequence H = (1, 4, a, . . .), a ≤ 7 satisfies the condition ∆H≤j/2 is an O-sequence (Theorem 4.2, 4.4). Using these results, they show that if H = (1, 4, 7, h, b, . . . , 1) is a Gorenstein sequence satisfying 3h− b− 17 ≥ 0, then the Zariski closure C(H) of the subscheme C(H) ⊂ PGor(H) parametrizing Artinian Gorenstein quotients A = R/I with I2 ∼= 〈wx,wy, wz〉 is a generically smooth component of PGor(H) (Theorem 4.6). They show that if in addition 8 ≤ h ≤ 10, then such PGor(H) have several irreducible components (Theorem 4.9). M. Boij and others had given previous examples of certain PGor(H) having several components in embedding dimension four or more [Bo2],[IK, Example C.38]. The proofs use properties of minimal resolutions, the smoothness of PGor(H ) for embedding dimension three [Klp], and the Gotzmann Hilbert scheme theorems [Go1, IKl].