Artinian Gorenstein algebras of embedding dimension four: components of [formula omitted] for [formula omitted

Abstract

A Gorenstein sequence H is a sequence of nonnegative integers H = ( 1 , h 1 , … , h j = 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 P Gor ( 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 h 1 ⩽ 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 I 2 ≅ 〈 wx , wy , wz 〉 (Theorems 3.7 and 3.9). They also show that any Gorenstein sequence H = ( 1 , 4 , a , … ) , a ⩽ 7 satisfies the condition Δ H ⩽ j / 2 is an O -sequence (Theorems 4.2 and 4.4). Using these results, they show that if H = ( 1 , 4 , 7 , h , b , … , 1 ) is a Gorenstein sequence satisfying 3 h - b - 17 ⩾ 0 , then the Zariski closure C ( H ) ¯ of the subscheme C ( H ) ⊂ P Gor ( H ) parametrizing Artinian Gorenstein quotients A = R / I with I 2 ≅ 〈 wx , wy , wz 〉 is a generically smooth component of P Gor ( H ) (Theorem 4.6). They show that if in addition 8 ⩽ h ⩽ 10 , then such P Gor ( H ) have several irreducible components (Theorem 4.9). M. Boij and others had given previous examples of certain P Gor ( H ) having several components in embedding dimension four or more (Pacific J. Math. 187(1) (1999) 1–11). The proofs use properties of minimal resolutions, the smoothness of P Gor ( H ′ ) for embedding dimension three (J.O. Kleppe, J. Algebra 200 (1998) 606–628), and the Gotzmann Hilbert scheme theorems (Math. Z. 158(1) (1978) 61–70).