Some generalizations of Castelnuovo’s lemma on zero-dimensional schemes.

*(English)*Zbl 0838.14040There 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)].

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)].

Reviewer: J.Migliore (Notre Dame)

##### MSC:

14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |

14H99 | Curves in algebraic geometry |