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Geometric consequences of extremal behavior in a theorem of Macaulay. (English) Zbl 0820.13019
Given positive integers \(i\) and \(h\), there is a unique expression \(h = {m_ i \choose i} + {m_{i-1} \choose i-1} + \cdots + {m_ j \choose j}\) with \(m_ i > m_{i-1} > \cdots > m_ j \geq j \geq 1\). Let \(h^{(i)} = {m_ j+ 1\choose i + 1} + {m_ i+1 \choose i} + \cdots + {m_ j+ 1\choose j + 1}\). Let \(A = k[X_ 1, \dots, X_ n]/I = \bigoplus A_ i\) be a graded algebra and let \(h_ i(A) = \dim_ k A_ i\). F. S. Macaulay [Proc. Lond. Math. Soc., II. Ser. 26, 531-555 (1927)] has shown that \((c_ 0, c_ 1, \dots) = (h_ 0(A), h_ 1(A), \dots)\) for some \(A\) if and only if \(c_ 0 = 1\) and \(c_{i+1} \leq c_ i^{(i)}\) for all \(i\). Suppose that for some \(d\): \(h_{d+1} = h_ d^{(d)}\). Then G. Gotzmann [Math. Z. 158, 61-70 (1978; Zbl 0358.13007)] showed that \(h_{i+1} = h_ i^{(i)}\) for any \(i \geq d\) if \(I\) has no generators of degree \(d\). The authors investigate some algebraic and geometric consequences arising from these theorems. The principal application is the study of Hilbert functions of zero-schemes with uniformity properties. As a consequence new limitations on the possible Hilbert functions of points arising from a general hyperplane section of an irreducible curve are obtained.
Example of an algebraic result: Let \(I\) be a homogeneous ideal in \(S = k[X_ 0, \dots, X_ n]\) with \(I_ d\neq 0\) and assume that \(S/I\) has maximal growth in degree \(d\). Then both \(I_ d\) and \(I_{d+1}\) have a GCD of “high” degree.
Example of a geometric result: Let \(Z\) be a reduced set of points in \(\mathbb{P}^{r+1}\), \(r + 1 \geq 3\). Assume that \(h_ d (I_ Z) - h_{d-1} (I_ Z) = h_{d+1} (I_ Z) - h_ d(I_ Z) = s\) for some \(d \geq s\). Then \(((I_ Z)_{\leq d}) = I_ C\) for some reduced curve \(C\) of degree \(s\). If the points of \(Z\) are in linear general position and \(r + 1 \leq s \leq 2(r + 1)\) then \(C\) is irreducible and “almost all” points lie on \(C\). If the points are in uniform position (no restriction on \(s)\) all points lie on \(C\).

MSC:
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14A15 Schemes and morphisms
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