Gale duality and free resolutions of ideals of points.

*(English)*Zbl 0943.13011How is the resolution of the ideal of a set of distinct generic points in \({\mathbb{P}}^n\) like? A conjecture about the graded Betti numbers of such resolutions (known as the “minimal resolution conjecture”, MRC) was given by A. Lorenzini [J. Algebra 156, No. 1, 5-35 (1993; Zbl 0811.13008)], and it has been proved in many cases and even asymptotically (when the number of points is much bigger then \(n\), by a result of Hirschowitz and Simpson).

Examples found computationally (Schreyer 1993) suggested nevertheless that the MRC could be false in general, even if no “geometrical reason” for those (three) counterexamples was known.

In the beautiful paper under review, such reason is found, enclosed in the theory of “Gale transforms” (they could be viewed as duality of linear series on a finite Gorenstein scheme), a way to associate to a set \(\Gamma \) of \(\gamma \geq r+3\) distinct points in \({\mathbb{P}}^r\) another set \(\Gamma ^\prime \) of \(\gamma \) points in \({\mathbb{P}}^s\), with \(s={\gamma -r -2}\).

The idea, expressed in classical language is this: With an appropriate choice of the coordinates, we can suppose that the coordinates of the points of \(\Gamma \) are the rows of a \((r+1)\times \gamma\) matrix \((I_{r+1}|B)\); then the coordinates of the points in \(\Gamma ^\prime \) are the rows of the matrix \((B^T|I_{s+1})\).

In the paper the relation among graded Betti numbers of \(\Gamma \) and of \(\Gamma ^\prime\) are found, and the “mystery” of the counterexamples to MRC is solved, moreover an infinite family of counterexample is determined (in any \({\mathbb{P}}^r\) with \(r\geq 6\), \(r\neq 9\)).

Examples found computationally (Schreyer 1993) suggested nevertheless that the MRC could be false in general, even if no “geometrical reason” for those (three) counterexamples was known.

In the beautiful paper under review, such reason is found, enclosed in the theory of “Gale transforms” (they could be viewed as duality of linear series on a finite Gorenstein scheme), a way to associate to a set \(\Gamma \) of \(\gamma \geq r+3\) distinct points in \({\mathbb{P}}^r\) another set \(\Gamma ^\prime \) of \(\gamma \) points in \({\mathbb{P}}^s\), with \(s={\gamma -r -2}\).

The idea, expressed in classical language is this: With an appropriate choice of the coordinates, we can suppose that the coordinates of the points of \(\Gamma \) are the rows of a \((r+1)\times \gamma\) matrix \((I_{r+1}|B)\); then the coordinates of the points in \(\Gamma ^\prime \) are the rows of the matrix \((B^T|I_{s+1})\).

In the paper the relation among graded Betti numbers of \(\Gamma \) and of \(\Gamma ^\prime\) are found, and the “mystery” of the counterexamples to MRC is solved, moreover an infinite family of counterexample is determined (in any \({\mathbb{P}}^r\) with \(r\geq 6\), \(r\neq 9\)).

Reviewer: Alessandro Gimigliano (Bologna)