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

Abstract

Star configurations of hypersurfaces are schemes in 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 is I (m) 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 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 assumption. Along the way, we exploit a connection between the minimal generators G (m) of I (m) and positive solutions to Diophantine equations, and a connection between G (m) and partitions of m via the canonical form of monomials. Our methods are characteristic-free.

Steiner Configurations ideals: containment and colouring

“…For other results on this topic we can also see [27,47]. Another tool useful to measure the non-containment among symbolic and ordinary powers of ideals is the notion of resurgence ρ(I) of an ideal I, introduced in [9] that gives some notion of how small the ratio m/r can be and still be sure to have I (m) ⊆ I r .…”

Symbolic powers and free resolutions of generalized star configurations of hypersurfaces

“…It has been of great interest to study various algebraic, geometric and combinatorial properties of star configurations; see for instance [3], [12], [13], [18] and the references therein. The object we are mostly interested here is the defining ideal…”

“…Consequently, these symbolic power ideals will have linear-like resolutions. This idea was later made more precise by Mantero in [18] by the notion of Koszul stranded Betti table; see Definition 1.2 below. Actually he showed that these ideals have complete intersection quotients.…”

“…Notice that with regard to general ideals lacking mutigrading structure, even for monomial ideals in the generic forms in F , weird phenomena emerge for the descriptions of associated primes, containment problems and the colon operations; see for example [18,Remark 3.2]. Therefore, one won't be surprised to see computations by the software Macaulay2 [14] showing that the three questions raised above have negative answers in general.…”

Betti numbers of symmetric shifted ideals

“…Shortly after this paper was posted on arXiv, another preprint appeared by Paolo Mantero [Man19] which also computes the graded Betti numbers of symbolic powers of star configurations. The results in Mantero's preprint were obtained independently from ours, and utilize new and interesting techniques.…”