From grids to pseudo-grids of lines: resolution and seminormality

TL;DR

This paper studies the minimal free resolutions of line configurations called complete grids and pseudo-grids over infinite fields, providing explicit descriptions, Betti number characterizations, and seminormality conditions, with new insights into their resolution maps.

Contribution

It explicitly describes minimal free resolutions of complete grids and pseudo-grids, characterizes their Betti numbers, and analyzes seminormality, offering new contributions on resolution maps.

Findings

Explicit minimal free resolutions for complete grids and pseudo-grids.

Characterization of total Betti numbers under multiplicity conditions.

Conditions for seminormality of pseudo-grids versus grids.

Abstract

Over an infinite field $K$, we investigate the minimal free resolution of some configurations of lines. We explicitly describe the minimal free resolution of "complete grids of lines" and obtain an analogous result about the so-called "complete pseudo-grids". Moreover, we characterize the total Betti numbers of configurations that are obtained posing a multiplicity condition on the lines of either a complete grid or a complete pseudo-grid. Finally, we analyze when a complete pseudo-grid is seminormal, differently from a complete grid. The main tools that have been involved in our study are the mapping cone procedure and properties of liftings, of pseudo-liftings and of weighted ideals. Although complete grids and pseudo-grids are hypersurface configurations and many results about such type of configurations have already been stated in literature, we give new contributions, in particular about the maps of the resolution.