Artinian Gorenstein algebras of embedding dimension four: components of image 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 image parametrizes all such Gorenstein algebra quotients of R having Hilbert function H and it is known to be smooth when the embedding dimension satisfies h1less-than-or-equals, slant3. 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 I2congruent withleft angle bracketwx,wy,wzright-pointing angle bracket (Theorems 3.7 and 3.9). They also show that any Gorenstein sequence H=(1,4,a,…),aless-than-or-equals, slant7 satisfies the condition ΔHless-than-or-equals, slantj/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 3h-b-17greater-or-equal, slanted0, then the Zariski closure image of the subscheme image parametrizing Artinian Gorenstein quotients A=R/I with I2congruent withleft angle bracketwx,wy,wzright-pointing angle bracket is a generically smooth component of image (Theorem 4.6). They show that if in addition 8less-than-or-equals, slanthless-than-or-equals, slant10, then such image have several irreducible components (Theorem 4.9). M. Boij and others had given previous examples of certain image 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 image 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).