On the vanishing topology of isolated Cohen–Macaulay codimension 2 singularities

Frühbis-Krüger, Anne¹ ; Zach, Matthias²

¹ Institut fur Mathematik, Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany
² Institut für Algebraische Geometrie, Leibniz Universität Hannover, Hannover, Germany

Geometry & topology, Tome 25 (2021) no. 5, pp. 2167-2194.

Résumé

We establish the rationality of simple isolated Cohen–Macaulay codimension $2$ (ICMC2) singularities in all dimensions $\ge 2$ and explicitly compute the vanishing homology of a certain class of threefolds including all the simple ones. ICMC2 singularities are determinantal and can be viewed as a natural generalization of complete intersections. The main tool for our investigations is the so-called Tjurina transformation — a special blowup construction based on the determinantal structure and often compatible with deformations.

Détail

DOI : 10.2140/gt.2021.25.2167

Classification : 32S30, 14B05
Keywords: singularity, Milnor fiber, vanishing homology, determinantal singularity

Bibliographie

[1] M Artin, Algebraic construction of Brieskorn’s resolutions, J. Algebra 29 (1974) 330 | DOI

[2] J Damon, B Pike, Solvable groups, free divisors and nonisolated matrix singularities, II : Vanishing topology, Geom. Topol. 18 (2014) 911 | DOI

[3] W Decker, G M Greuel, G Pfister, H Schönemann, Singular: a computer algebra system for polynomial computations (2019)

[4] D Eisenbud, Commutative algebra: with a view toward algebraic geometry, 150, Springer (1995) | DOI

[5] A Frühbis-Krüger, Classification of simple space curve singularities, Comm. Algebra 27 (1999) 3993 | DOI

[6] A Frühbis-Krüger, On discriminants, Tjurina modifications and the geometry of determinantal singularities, Topol. Appl. 234 (2018) 375 | DOI

[7] A Frühbis-Krüger, A Neumer, Simple Cohen–Macaulay codimension 2 singularities, Comm. Algebra 38 (2010) 454 | DOI

[8] T Gaffney, A Rangachev, Pairs of modules and determinantal isolated singularities, preprint (2014)

[9] S G Gindikin, G M Khenkin, editors, Several complex variables, IV : Algebraic aspects of complex analysis, 10, Springer (1990) | DOI

[10] M Giusti, Classification des singularités isolées simples d’intersections complètes, from: "Singularities, I" (editor P Orlik), Proc. Sympos. Pure Math. 40, Amer. Math. Soc. (1983) 457 | DOI

[11] V Goryunov, D Mond, Tjurina and Milnor numbers of matrix singularities, J. Lond. Math. Soc. 72 (2005) 205 | DOI

[12] H Grauert, Über die Deformation isolierter Singularitäten analytischer Mengen, Invent. Math. 15 (1972) 171 | DOI

[13] H Grauert, T Peternell, R Remmert, editors, Several complex variables, VII : Sheaf-theoretical methods in complex analysis, 74, Springer (1994) | DOI

[14] G M Greuel, J Steenbrink, On the topology of smoothable singularities, from: "Singularities, I" (editor P Orlik), Proc. Sympos. Pure Math. 40, Amer. Math. Soc. (1983) 535 | DOI

[15] S M Gusein-Zade, W Ebeling, On the indices of 1–forms on determinantal singularities, Tr. Mat. Inst. Steklova 267 (2009) 119

[16] H Hamm, Lokale topologische Eigenschaften komplexer Räume, Math. Ann. 191 (1971) 235 | DOI

[17] R Hartshorne, Deformation theory, 257, Springer (2010) | DOI

[18] S Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18 (1964) 449

[19] E Looijenga, J Steenbrink, Milnor number and Tjurina number of complete intersections, Math. Ann. 271 (1985) 121 | DOI

[20] J Milnor, Singular points of complex hypersurfaces, 61, Princeton Univ. Press (1968)

[21] H Møller Pedersen, On Tjurina transform and resolution of determinantal singularities, preprint (2016)

[22] D Mond, D Van Straten, Milnor number equals Tjurina number for functions on space curves, J. Lond. Math. Soc. 63 (2001) 177 | DOI

[23] T Okuma, Numerical Gorenstein elliptic singularities, Math. Z. 249 (2005) 31 | DOI

[24] O Riemenschneider, Familien komplexer Räume mit streng pseudokonvexer spezieller Faser, Comment. Math. Helv. 51 (1976) 547 | DOI

[25] K Saito, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math. 14 (1971) 123 | DOI

[26] M Schaps, Nonsingular deformations of a determinantal scheme, Pacific J. Math. 65 (1976) 209 | DOI

[27] M Schaps, Deformations of Cohen–Macaulay schemes of codimension 2 and non-singular deformations of space curves, Amer. J. Math. 99 (1977) 669 | DOI

[28] D Van Straten, Weakly normal surface singularities and their improvements, PhD thesis, Universiteit Leiden (1987)

[29] G N Tyurina, Absolute isolatedness of rational singularities and triple rational points, Funkcional. Anal. i Priložen. 2 (1968) 70

[30] J M Wahl, Equations defining rational singularities, Ann. Sci. École Norm. Sup. 10 (1977) 231 | DOI

[31] J M Wahl, Simultaneous resolution of rational singularities, Compos. Math. 38 (1979) 43

[32] J Wahl, Milnor and Tjurina numbers for smoothings of surface singularities, Algebr. Geom. 2 (2015) 315 | DOI

[33] M Zach, An observation concerning the vanishing topology of certain isolated determinantal singularities, Math. Z. 291 (2019) 1263 | DOI