Generalized star configurations and the Tutte polynomial

TL;DR

This paper explores the relationship between generalized star configurations derived from linear codes and the Tutte polynomial, revealing how the degree of these configurations is determined by the polynomial.

Contribution

It establishes a connection between the degree of generalized star configurations and the Tutte polynomial of the code, and investigates their algebraic properties.

Findings

Degree of generalized star configurations is determined by the Tutte polynomial.

Preliminary results on the primary decomposition of defining ideals.

Partial proof supporting the conjecture of linear minimal free resolutions.

Abstract

From the generating matrix of a linear code one can construct a sequence of generalized star configurations which are strongly connected to the generalized Hamming weights and the underlying matroid of the code. When the code is MDS, the matrix is generic and we obtain the usual star configurations. In our main result, we show that the degree of a generalized star configuration as a projective scheme is determined by the Tutte polynomial of the code. In the process, we obtain preliminary results on the primary decomposition of the defining ideals of these schemes. Additionally, we conjecture that these ideals have linear minimal free resolutions and prove partial results in this direction.