1. Distinguishing $\Bbbk$-configurations
- Author
-
Yong-Su Shin, Adam Van Tuyl, and Federico Galetto
- Subjects
Hilbert series and Hilbert polynomial ,Sequence ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Multiplicity (mathematics) ,Type (model theory) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,13D40 ,01 natural sciences ,Combinatorics ,Mathematics - Algebraic Geometry ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,symbols ,13D40, 14M05 ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,14M05 ,Mathematics - Abstract
A $\Bbbk$-configuration is a set of points $\mathbb{X}$ in $\mathbb{P}^2$ that satisfies a number of geometric conditions. Associated to a $\Bbbk$-configuration is a sequence $(d_1,\ldots,d_s)$ of positive integers, called its type, which encodes many of its homological invariants. We distinguish $\Bbbk$-configurations by counting the number of lines that contain $d_s$ points of $\mathbb{X}$. In particular, we show that for all integers $m \gg 0$, the number of such lines is precisely the value of $\Delta \mathbf{H}_{m\mathbb{X}}(m d_s -1)$. Here, $\Delta \mathbf{H}_{m\mathbb{X}}(-)$ is the first difference of the Hilbert function of the fat points of multiplicity $m$ supported on $\mathbb{X}$., Comment: Revised version of paper; most changes minor except the proof of Lemma 4.1 which has been rewritten; to appear in Illinois Journal of Mathematics
- Published
- 2017