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Some generalizations of Castelnuovo’s lemma on zero-dimensional schemes. (English) Zbl 0838.14040
There are two main parts of this paper. The unifying theme is that the algebraic structure of the canonical module $$\omega_R$$ of the coordinate ring $$R$$ of a zero-dimensional subscheme $$X$$ of projective space determines whether or not $$X$$ lies on a rational normal curve (resp. a rational normal scroll).
In the first part, the author gives a nice extension of some results of D. Eisenbud and J. Harris [J. Algebr. Geom. 1, No. 1, 15-30 (1992; Zbl 0804.14002) and 31-59 (1992; Zbl 0798.14029)] on Castelnuovo’s lemma. Given a zeroscheme $$X$$ (not necessarily reduced) of degree $$\geq 2r + 2 + d$$ in $$\mathbb{P}^r$$ in uniform position, which imposes only $$2r + d$$ conditions on quadrics, the author shows that there is a $$d$$-dimensional rational normal scroll containing $$X$$. The case $$d = 1$$ was done by Eisenbud and Harris, who used it to study “nearly extremal” curves with respect to Castelnuovo theory. They also showed that if deg $$X = r + 3$$ then $$X$$ lies on a unique rational normal curve.
In the second half of the paper, the author examines the range $$r + 4 \leq \deg X \leq 2r + 2$$, which is the range not covered in the results of Eisenbud and Harris mentioned above. He looks at the matrix corresponding to the multiplication $$(\omega_R)_{-1} \otimes S_1 \to (\omega_R )_0$$ (where $$S$$ is the polynomial ring), and shows that this matrix is equivalent to the Hankel matrix if and only if $$X$$ lies on a unique rational normal curve. He also gives some results on the graded Betti numbers of $$X$$ in this case. The main tools used are some results of M. Kreuzer on canonical modules [Can. J. Math. 46, No. 2, 357-379 (1994; Zbl 0826.14030)], and work of D. Eisenbud [Am. J. Math. 110, No. 3, 541-575 (1988; Zbl 0681.14028)] on 1-generic matrices. The author has recently given a very nice generalization of his work under review here [J. Pure Appl. Algebra 105, No. 1, 107-116 (1995)].

##### MSC:
 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14H99 Curves in algebraic geometry
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