Abstract
We discuss how to resolve generic skew-symmetric and generic symmetric determinantal singularities. The key ingredients are (skew-) symmetry preserving matrix operations in order to deduce an inductive argument.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bruce, J. W., Goryunov, V. V., Haslinger, G. J.: Families of skew-symmetric matrices of even size, (2022). Preprint available on arXiv:2206.00596
Bierstone, Edward, Milman, Pierre D.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997)
Bruns, Winfried: Lecture Notes in Mathematics Vetter, Udo: Determinantal rings, p. 1327. Springer-Verlag, Berlin (1988)
Steven Dale Cutkosky: Resolution of singularities. Graduate Studies in Mathematics, vol. 63. American Mathematical Society, Providence, RI (2004)
Encinas, Santiago, Villamayor, Orlando: A course on constructive desingularization and equivariance. Resolution of singularities (Obergurgl, 1997), 147–227, Progr. Math., 181, Birkhäuser, Basel, (2000)
Frühbis-Krüger, Anne, Zach, Matthias: Determinantal singularities. Handbook of geometry and topology of singularities IV, 45–159, Springer, Cham, (2023)
Gaube, Sabrina Alexandra: Specialized strategies for resolution of singularities of determinantal ideals. PhD thesis, Carl von Ossietzky Universität Oldenburg, (2023)
Gaube, Sabrina Alexandra: Resolution of determinantal ideals with exploition of the matrix structure. In preparation
Gaffney, T., Molino, M.: Symmetric determinantal singularities I: the multiplicity of the polar curve, (2020)
Gaffney, T., Molino, M.: Symmetric determinantal singularities II: equisingularity and seids, (2021)
Harris, J.: Algebraic geometry, volume 133 of Graduate Texts in Mathematics. Springer-Verlag, New York, (1992). A first course
Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero I, II. Ann. Math. 2(79), 109–203 (1964)
Kodiyalam, Vijay, Lam, T. Y., Swan, R. G.: Determinantal ideals, Pfaffian ideals, and the principal minor theorem. In Noncommutative rings, group rings, diagram algebras and their applications, volume 456 of Contemp. Math., pp 35–60. Amer. Math. Soc., Providence, RI, (2008)
Kollár, János.: Lectures on resolution of singularities. Annals of Mathematics Studies, vol. 166. Princeton University Press, Princeton, NJ (2007)
Schober, Bernd: Partial local resolution by characteristic zero methods. Results Math. 73(1), 39–48 (2018)
Vainsencher, Israel: Complete collineations and blowing up determinantal ideals. Math. Ann. 267(3), 417–432 (1984)
Villamayor, Orlando: Constructiveness of Hironaka’s resolution. Ann. Sci. École Norm. Sup. (4) 22(1), 1–32 (1989)
Orlando, E., Villamayor, U.: Patching local uniformizations. Ann. Sci. École Norm. Sup. 25(6), 629–677 (1992)
Acknowledgements
Bernd Schober thanks Laura Escobar for joint discussions on (skew-) symmetry preserving decompositions of matrices in the context of a different topic which inspired the methods for reduction in the present article. Both authors thank the referees for useful comments on an earlier version of the article and in particular, for pointing out parts that needed clarification in the proof of Theorem A.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Data Availibility Statement
Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
B.S. was partially supported by the project “Order zeta functions and resolutions of singularities” funded by the Deutsche Forschungsgemeinschaft (DFG) (DFG Project Number: 373111162).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Gaube, S.A., Schober, B. Desingularization of generic symmetric and generic skew-symmetric determinantal singularities. manuscripta math. (2024). https://doi.org/10.1007/s00229-024-01544-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00229-024-01544-4