Ideals generated by $a$-fold products of linear forms have linear graded free resolution
Abstract
Given $\Sigma\subset R:=\mathbb K[x_1,\ldots,x_k]$, where $\mathbb K$ is a field of characteristic 0, any finite collection of linear forms, some possibly proportional, and any $1\leq a\leq |\Sigma|$, we prove that $I_a(\Sigma)$, the ideal generated by all $a$-fold products of $\Sigma$, has linear graded free resolution. This allows us to determine a generating set for the defining ideal of the Orlik-Terao algebra of the second order of a line arrangement in $\mathbb P_{\mathbb{K}}^2$, and to conclude that for the case $k=3$, and $\Sigma$ defining such a line arrangement, the ideal $I_{|\Sigma|-2}(\Sigma)$ is of fiber type. We also prove several conjectures of symbolic powers for defining ideals of star configurations of any codimension $c$.
- Publication:
-
arXiv e-prints
- Pub Date:
- April 2020
- DOI:
- 10.48550/arXiv.2004.07430
- arXiv:
- arXiv:2004.07430
- Bibcode:
- 2020arXiv200407430B
- Keywords:
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- Mathematics - Commutative Algebra;
- Mathematics - Algebraic Geometry;
- 13D02 (Primary) 14N20;
- 52C35;
- 13A30;
- 14Q99 (Secondary)
- E-Print:
- 16 pages. In this new version, with appropriate new title, we prove in its full generality the conjecture that any ideal generated by all a-fold products of linear forms has linear graded free resolution. We also prove various conjectures that involve symbolic powers of star configurations of any codimension