On jet closures of singularities

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Аннотация

Jet closure and jet support closure were first introduced by de Fernex, Ein and Ishii to solve the local isomorphism problem. In this paper we introduce two local algebras associated to jet closure and jet support closure, respectively. We show that these two algebras are invariants of singularities. We compute and investigate these invariants for some interesting cases, such as the cases of monomial ideals and homogeneous ideals. For application, we can distinguish different simple curve singularities by a finite number of jet support closures, and this number is close to the Milnor number of the singularity. We also introduce a new filtration and a jet index for jet closures. The jet index describes which jet scheme recovers the information on the base scheme. Moreover, we obtain some properties of the jet index. Bibliography: 16 titles.

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Авторлар туралы

Yifan Chen

Department of Mathematical Sciences, Tsinghua University, Beijing, P. R. China

Хат алмасуға жауапты Автор.
Email: c-yf20@tsinghua.org.cn

Huaiqing Zuo

Department of Mathematical Sciences, Tsinghua University, Beijing, P. R. China

Email: hqzuo@mail.tsinghua.edu.cn

Әдебиет тізімі

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  3. T. de Fernex, L. Ein, S. Ishii, “Jet closures and the local isomorphism problem”, J. Algebra, 501 (2018), 166–181
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  9. S. Ishii, “Finite determination conjecture for Mather–Jacobian minimal log discrepancies and its applications”, Eur. J. Math., 4:4 (2018), 1433–1475
  10. S. Ishii, A. J. Reguera, “Singularities with the highest Mather minimal log discrepancy”, Math. Z., 275:3-4 (2013), 1255–1274
  11. S. Ishii, A. J. Reguera, “Singularities in arbitrary characteristic via jet schemes”, Hodge theory and $L^2$-analysis, Adv. Lect. Math. (ALM), 39, Int. Press, Somerville, MA, 2017, 419–449
  12. D. Rees, Lectures on the asymptotic theory of ideals, London Math. Soc. Lecture Note Ser., 113, Cambridge Univ. Press, Cambridge, 1988, x+202 pp.
  13. J. F. Nash, Jr., “Arc structure of singularities”, Duke Math. J., 81:1 (1995), 31–38
  14. J. Denef, F. Loeser, “Germs of arcs on singular algebraic varieties and motivic integration”, Invent. Math., 135:1 (1999), 201–232
  15. M. Mustaţǎ, “Singularities of pairs via jet schemes”, J. Amer. Math. Soc., 15:3 (2002), 599–615
  16. L. Ein, M. Mustaţǎ, “Jet schemes and singularities”, Algebraic geometry–Seattle 2005, Part 2, Proc. Sympos. Pure Math., 80, Part 2, Amer. Math. Soc., Providence, RI, 2009, 505–546

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© Chen Y., Zuo H., 2025

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