The structure and free resolutions of the symbolic powers of star configurations of hypersurfaces

Mantero, Paolo

doi:10.1090/tran/8208

The structure and free resolutions of the symbolic powers of star configurations of hypersurfaces
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Trans. Amer. Math. Soc. 373 (2020), 8785-8835 Request permission

Abstract:

Star configurations of hypersurfaces are schemes in $\mathbb P^n$ widely generalizing star configurations of points. Their rich structure allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra, and representation theory. In particular, there has been much interest in understanding how “fattening" these schemes affects the algebraic properties of these configurations or, in other words, understanding the symbolic powers $I^{(m)}$ of their defining ideals $I$.

In the present paper (1) we prove a structure theorem for $I^{(m)}$, giving an explicit description of a minimal generating set of $I^{(m)}$ (overall, and in each degree) which also yields a minimal generating set of the module $I^{(m)}/I^m$—which measures how far $I^{(m)}$ is from $I^m$. These results are new even for monomial star configurations or star configurations of points; (2) we introduce a notion of ideals with c.i. quotients, generalizing ideals with linear quotients, and show that $I^{(m)}$ have c.i. quotients. As a corollary we obtain that symbolic powers of ideals of star configurations of points have linear quotients; (3) we find a general formula for all graded Betti numbers of $I^{(m)}$; (4) we prove that a little bit more than the bottom half of the Betti table of $I^{(m)}$ has a regular, almost hypnotic, pattern, and we provide a simple closed formula for all these graded Betti numbers and the last irregular strand in the Betti table.

Other applications include improving and widely extending results by Galetto, Geramita, Shin, and Van Tuyl, and providing explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations.

Inspired by Young tableaux, we introduce a “canonical" way of writing any monomial in any given set of polynomials, which may be of independent interest. We prove its existence and uniqueness under fairly general assumptions. Along the way, we exploit a connection between the minimal generators $\mathbf {G}_{(m)}$ of $I^{(m)}$ and positive solutions to Diophantine equations, and a connection between $\mathbf {G}_{(m)}$ and partitions of $m$ via the canonical form of monomials. Our methods are characteristic-free.

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