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By Eben Matlis
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Additional info for 1-Dimensional Cohen-Macaulay Rings
Elements in PGL2 C commuting with ω must be of the form a b −b a or a b b −a , and if these commute also with ρ, the elements 1, ω, ρ, ωρ are the only possibilities. 3. The variety (V (4)H )0 ⊂ V (4) consisting of those points whose stabilizer in PSL2 C is exactly H is a (PSL2 C, N (H))-section of V (4). Proof. The fact that the orbit PSL2 C · (V (4)H )0 is dense in V (4) follows since a general point in V (4) has stabilizer conjugate to H; the assertion ∀g ∈ PGL2 C, ∀x ∈ (V (4)H )0 : gx ∈ (V (4)H )0 =⇒ g ∈ N (H) is clear by deﬁnition.
M (um )) = δ 1 δ 2 . . δ m ◦ U m (um ) = cm vm so we set m := πa,b 1 m L . cm (58) m−l m−l+1 m , πa,b , . . , πa,b have already been Now assume, by induction, that πa,b m−l−1 determined. We show how to calculate πa,b . Now, by (54), vm−l−1 ∈ δ 1 . . δ m−l−1 (ker Δm−l−1 ). We write vm−l−1 = 1 δ . . δ m−l−1 (um−l−1 ), for some um−l−1 ∈ ker Δm−l−1 = V m−l−1 (a − (m − l − 1), b − (m − l − 1)), and using (57) we get l Lm−l−1 = v− m−i πa,b (v) i=0 m−l−1 L (vm−l−1 ) = Lm−l−1 (v0 + v1 + · · · + vm−l−1 ) = Lm−l−1 (δ 1 .
Y12 ); the coeﬃcients of the monomials in the y’s are (inhomogeneous) polynomials of degrees ≤ 2 in r1 , r2 , r3 . For r1 = r2 = r3 = 0, q1 , q2 do not vanish identically. (2) The polynomials q3 , q4 , q5 are homogeneous linear in (y1 , . . , y12 ); the coeﬃcients of the monomials in the y’s are (inhomogeneous) polynomials of degrees ≤ 2 in r1 , r2 , r3 . For r1 = r2 = r3 = 0, q3 , q4 , q5 do not vanish identically. 1. Let Y˜λ be the subvariety of R × P8 deﬁned by the equations q1 = q2 = q3 = q4 = q5 = 0.
1-Dimensional Cohen-Macaulay Rings by Eben Matlis