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Matroid connectivity and singularities of configuration hypersurfaces
Graham Denham1, Mathias Schulze2, Uli Walther3
1Department of Mathematics, University of Western Ontario, London, ON N6A 5B7 Canada.
Letters in Mathematical Physics
|February 11, 2021
Summary
This study investigates degeneracy schemes of bilinear forms associated with linear matroids. We demonstrate these schemes are reduced and analyze how matroid connectivity impacts their properties, such as integrality.
Area of Science:
- Algebraic Geometry
- Combinatorics
- Commutative Algebra
Background:
- Linear realizations of matroids over fields are fundamental in combinatorial abstract algebra.
- Associated configuration polynomials and symmetric bilinear forms with linear homogeneous coefficients are key objects of study.
- Degeneracy schemes of these forms, specifically the first and second, are linked to the configuration hypersurface and its non-smooth locus.
Purpose of the Study:
- To demonstrate that the first and second degeneracy schemes of the bilinear form are reduced.
- To elucidate the impact of matroid connectivity on the properties of these degeneracy schemes.
- To describe the behavior of configuration polynomials, forms, and schemes under various matroid constructions.
Main Methods:
- Analysis of linear realizations of matroids.
- Construction and study of configuration polynomials and associated symmetric bilinear forms.
- Investigation of degeneracy schemes and their properties (reduced, integral, Cohen-Macaulay) based on matroid connectivity.
Main Results:
- The configuration hypersurface and its non-smooth locus support reduced first and second degeneracy schemes, respectively.
- For 2-connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen-Macaulay of codimension 3.
- For 3-connected matroids, the second degeneracy scheme is also integral.
Conclusions:
- Matroid connectivity plays a crucial role in determining the geometric and algebraic properties of degeneracy schemes.
- The study provides a detailed understanding of how these schemes behave under different matroid operations.
- Established results on reducedness, integrality, and Cohen-Macaulay properties offer significant insights into the structure of these algebraic-combinatorial objects.
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