What question did this study set out to answer?

To establish arithmetic criteria for identifying non-very generic intersections in discriminantal hyperplane arrangements.

March 21, 2026Open Access

Arithmetic non-very generic arrangements

PDPragnya DasIndian Institute of Information Technology Senapati, Manipur TSTakuya Saito SSSimona SettepanellaHokkaido University of Science

Key Points

To establish arithmetic criteria for identifying non-very generic intersections in discriminantal hyperplane arrangements.
Constructing discriminantal hyperplane arrangements from a generic hyperplane arrangement
Classifying the arrangement based on combinatorial structure
Analyzing intersection lattices to determine rank conditions
Identified specific criteria for recognizing non-very generic intersections
Provided corrections to previous research on rank 2 intersections
Demonstrated the implications of these findings on the structure of discriminantal arrangements

Abstract

Abstract A discriminantal hyperplane arrangement B (n, k, A) B (n, k, A) is constructed from a given generic hyperplane arrangement A A. The arrangement A A is classified as either very generic or non-very generic according to the combinatorial structure of B (n, k, A) B (n, k, A). In particular, A A is regarded as non-very generic if the intersection lattice of B (n, k, A) B (n, k, A) contains at least one non-very generic intersection, namely, an intersection that does not satisfy a specific rank condition established by Athanasiadis. In this paper, we present arithmetic criteria characterizing non-very generic intersections in discriminantal arrangements, and we complete and correct a previous result of Libgober and the third author concerning rank 2 intersections in such arrangements.

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