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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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