Classifying families of Gorenstein quotients of e.g.
codimension 4 of a polynomial ring
by Jan O. Kleppe
Abstract
Let
B
be a graded Cohen-Macaulay quotient of a polynomial ring
with Hilbert function
HB .
R
Firstly we need to recall some concepts such
as Cohen-Macaulay, Gorenstein, complete intersection and licci
and the denition and some properties of algebra cohomology. More-
B
A provided B → A is a given surjection.
over we need a Theorem of how deformations of a graded algebra
are related to deformations of
I recall this by some slides and I will give you a copy.
If KB , the canonical module of B , is locally free in some open subset
∗
Proj(B), one knows that a regular section σ of the B -dual KB
(s) (s
an integer) denes a Gorenstein quotient A given by the exact sequence
of
σ
0 → KB (−s) → B → A → 0 .
(1)
MB is a (locally free) maximal Cohen-Macaulay mod1 ≤ r ≤ 3 whose top exterior power is locally a twist of
KB , then a regular section σ of MB∗ (s) denes a Gorenstein algebra A
by (2) : A = B/ im σ , and we have dim A = dim B − r .
H
Let GradAlg (R) be the scheme parametrizing graded quotients of
R with Hilbert function H (this is essentially the usual Hilbert scheme
if deg p > 0, where p is the Hilbert polynomial; p(v) = H(v) for v >>
0). If WB ⊂ GradAlgHB (R) is an irreducible component containing an
HA
open subset U of quotients (B) as above, we let WA ⊂ GradAlg
(R)
be the closure of the locus of points (A) constructed by (1) or (2), by
varying (B) ∈ U and σ . Under certain assumptions, notably; s >> 0
(which we can make precise) or B in the linkage class of a complete
More generally if
ule of rank
intersection (in case (1), the case (2) requires more), we are able to
determine
(i) the dimension of
WA
in terms of
WB
and
∗
dim(KB
)s
and a well
δ,
HA
(ii) whether WA is an irreducible component of GradAlg
(R) (this
happens essentially when δ = 0), and
HA
(iii) when GradAlg
(R) is generically smooth along WA .
In fact for s >> 0 there is a well dened injective application from
HB
the set of irreducible components of GradAlg
(R) to the set of irHA
reducible components of GradAlg
(R). In the applications we focus
on algebras A of codimension 4 of R.
dened invariant
1