Abstract:
We provide a criterion that for an equivalence group G on holomorphic germs, the discriminant of a G-versal unfolding is a free divisor. The criterion is in terms of the discriminant being Cohen–Macaulay and generically having Morse-type singularities. When either of these conditions fails, we provide a criterion that the discriminant have a weaker free* divisor structure. For nonlinear sections of a free* divisor V, we obtain a formula for the number of singular vanishing cycles by modifying an earlier formula obtained with David Mond and taking into account virtual singularities.
Key words and phrases:
Discriminants, versal unfoldings, free divisors, free* divisors, liftable vector fields, Morse-type singularities, Cohen–Macaulay condition.
\Bibitem{Dam03}
\by J.~Damon
\paper On the legacy of free divisors. II.~Free* divisors and complete intersections
\jour Mosc. Math.~J.
\yr 2003
\vol 3
\issue 2
\pages 361--395
\mathnet{http://mi.mathnet.ru/mmj91}
\crossref{https://doi.org/10.17323/1609-4514-2003-3-2-361-395}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2025265}
\zmath{https://zbmath.org/?q=an:1040.32026}
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Linking options:
https://www.mathnet.ru/eng/mmj91
https://www.mathnet.ru/eng/mmj/v3/i2/p361
This publication is cited in the following 7 articles:
Shinichi Tajima, Takafumi Shibuta, Katsusuke Nabeshima, Lecture Notes in Computer Science, 12291, Computer Algebra in Scientific Computing, 2020, 543
Nabeshima K., Tajima Sh., “Computation Methods of Logarithmic Vector Fields Associated to Semi-Weighted Homogeneous Isolated Hypersurface Singularities”, Tsukuba J. Math., 42:2 (2018), 191–231
Damon J. Pike B., “Solvable Groups, Free Divisors and Nonisolated Matrix Singularities i: Towers of Free Divisors”, Ann. Inst. Fourier, 65:3 (2015), 1251–1300
Damon J., Pike B., “Solvable group representations and free divisors whose complements are K(pi, 1)'s”, Topology Appl, 159:2 (2012), 437–449
Buchweitz R.-O., Ebeling W., von Bothmer H.-Ch.G., “Low-Dimensional Singularities with Free Divisors as Discriminants”, Journal of Algebraic Geometry, 18:2 (2009), 371–406
Damon J., “On the legacy of free divisors III: Functions and divisors on complete intersections”, Quarterly Journal of Mathematics, 57:1 (2006), 49–79
Anne Frühbis-Krüger, “Partial standard bases as a tool for studying families of singularities”, Journal of Symbolic Computation, 38:4 (2004), 1191
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