The paper [10] raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p ₁+...+2p _r, for all possible choices ofr distinct points in ℙ². We study this problem forr points in ℙ² over an algebraically closed fieldk of arbitrary characteristic in case eitherr ≤ 8 or the points lie on a (possibly reducible) conic. In either case, it follows from [17, 18] that there are only finitely many configuration types of points, where our notion of configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We sayp ₁, ...,p _r andp ₁ ^′, ... ,p _r ^′ have the sameconfiguration type if for all choices of nonnegative integersm _i ,Z=m ₁ p ₁+...+m _r p _r andZ′=m ₁ p ₁ ^′+...+m _r p _r ^′ have the same Hilbert function.) Assuming either that 7 ≤r ≤ 8 (see [12] for the casesr ≤6) or that the pointsp _i lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficientsm _i determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) ofZ=m ₁ p ₁+...+m _r p _r . We demonstrate our results by explicitly listing all Hilbert functions for schemes ofr≤ 8 double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function.

中文翻译：

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ℙ2中肥胖点子方案的希尔伯特函数分类

文献[10]中提出的什么可能的希尔伯特函数是形式2的脂肪点subschemes问题p ₁ + ... + 2 p _{- [R} ，对于所有可能的选择- [R在ℙ不同的点²。我们研究这个问题[R在ℙ分²比代数闭场ķ任意特征的情况下，或者[R≤8或这些点位于（可能是可还原的）圆锥上。无论哪种情况，从[17，18]中得出，只有有限多种点的配置类型，其中我们的配置类型概念是可表示的组合几何体（也称为可表示的简单拟阵线）概念的概括。（我们说p ₁，...，p _[R和p ₁ ^'，...，p _[R ^'具有相同的构型，如果对非负整数的所有选择中号 _我，ž =米₁ p ₁ + ... +米_{- [R} p _{- [R} 和ž '=米 ₁ p ₁ ^{' +} ... +米 _{- [R} p _{- [R} ^'具有相同的希尔伯特功能。）假设要么7≤ [R ≤8（见[12]的情况下- [R ≤6）或点p _i位于圆锥上，我们显式确定所有配置类型，并显示配置类型和系数m _i如何（以显式可计算的方式）确定Z = m ₁的希尔伯特函数（有时是分级的Betti数）p ₁ + ... + m _r p _r 。我们通过明确列出所有希尔伯特功能的方案展示我们的研究结果[R ≤8个双倍积分，并为每个希尔伯特功能，我们恰恰说明该点必须如何排列（配置类型方面），以获得希尔伯特功能。