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Distinguishing $\Bbbk$-configurations
- Publication Year :
- 2017
-
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 $\mathbb{X}$.<br />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
- Subjects :
- Mathematics - Commutative Algebra
Mathematics - Algebraic Geometry
13D40, 14M05
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1705.09195
- Document Type :
- Working Paper