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DEGENERATIONS OF ORBIFOLD CURVES AS NONCOMMUTATIVE VARIETIES

Published online by Cambridge University Press:  15 July 2025

SHINNOSUKE OKAWA*
Affiliation:
Department of Mathematics Graduate School of Science Osaka University Machikaneyama 1-1 Toyonaka Osaka, 560-0043 Japan
TARIG ABDELGADIR
Affiliation:
Mathematical Sciences https://ror.org/04vg4w365Loughborough University LE11 3TU United Kingdom t.abdelgadir@lboro.ac.uk
DANIEL CHAN
Affiliation:
School of Mathematics and Statistics https://ror.org/03r8z3t63UNSW Sydney NSW 2052 Australia danielc@unsw.edu.au
KAZUSHI UEDA
Affiliation:
Graduate School of Mathematical Sciences https://ror.org/057zh3y96The University of Tokyo 3-8-1 Komaba, Meguro-ku Tokyo, 153-8914 Japan kazushi@ms.u-tokyo.ac.jp

Abstract

Boundary points on the moduli space of pointed curves corresponding to collisions of marked points have modular interpretations as degenerate curves. In this paper, we study degenerations of orbifold projective curves corresponding to collisions of stacky points from the point of view of noncommutative algebraic geometry.

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Article
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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References

Abdelgadir, T., Chan, D., Okawa, S. and Ueda, K., A compact moduli of orbifold projective curves, preprint, arXiv:2401.14797.Google Scholar
Abdelgadir, T., Okawa, S. and Ueda, K., Compact moduli of non-commutative cubic surfaces, preprint, arXiv:2404.00175.Google Scholar
Abdelgadir, T., Okawa, S. and Ueda, K., Compact moduli of noncommutative projective planes, preprint, arXiv:1411.7770.Google Scholar
Abdelgadir, T. and Ueda, K., Weighted projective lines as fine moduli spaces of quiver representations , Commun. Algebra 43 (2015), 636649.10.1080/00927872.2013.842245CrossRefGoogle Scholar
Artin, M. and Zhang, J., Noncommutative projective schemes , Adv. Math. 109 (1994), 228287.10.1006/aima.1994.1087CrossRefGoogle Scholar
Bergman, G., The diamond lemma for ring theory , Adv. Math. 29 (1978), 178218.CrossRefGoogle Scholar
Bondal, A. I. and Polishchuk, A. E., Homological properties of associative algebras: the method of helices , Izv. Ross. Akad. Nauk Ser. Mat. 57 (1993), 350.Google Scholar
Crawley-Boevey, W., Noncommutative algebra 1, course notes available here Google Scholar
Geigle, W. and Lenzing, H., “A class of weighted projective curves arising in representation theory of finite-dimensional algebras” in G.-M. Greuel and G. Trautmann (eds.), Singularities, Representation of Algebras, and Vector Bundles (Lambrecht, 1985), Lecture Notes in Mathematics, 1273, Springer, Berlin, 1987.Google Scholar
Gelfand, I. M. and MacPherson, R. M., Geometry in Grassmannians and a generalization of the dilogarithm , Adv. Math. 44 (1982), 279312.10.1016/0001-8708(82)90040-8CrossRefGoogle Scholar
Mumford, D., Fogarty, J. and Kirwan, F. W., Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, Springer-Verlag, Berlin, 1994.10.1007/978-3-642-57916-5CrossRefGoogle Scholar
Polishchuk, A., Noncommutative proj and coherent algebras , Math. Res. Lett. 1 (1994), 6374.Google Scholar
The Stacks Project Authors, Stacks project, (2018). https://stacks.math.columbia.edu Google Scholar
Van den Bergh, M., Blowing up of non-commutative smooth surfaces , Mem. Amer. Math. Soc. 734 (2001), x+140.Google Scholar