Versality, bounds of global Tjurina numbers and logarithmic vector fields along hypersurfaces with isolated singularities. (English) Zbl 1468.14014
Fernández de Bobadilla, Javier (ed.) et al., Singularities and their interaction with geometry and low dimensional topology. In honor of András Némethi on the occasion of his 60th birthday. Selected papers based on the presentations at the conference “Némethi60: geometry and topology of singularities”, Budapest, Hungary, May 27–31, 2019. Basel: Birkhäuser/Springer. Trends Math., 1-12 (2021).
Summary: We recall first the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface \(V\) with isolated singularities and the versality properties of \(V\), as studied by A. A. Du Plessis and C. T. C. Wall [Math. Proc. Camb. Philos. Soc. 126, No. 2, 259–266 (1999; Zbl 0926.14012); J. Algebr. Geom. 9, No. 2, 309–322 (2000; Zbl 1006.14011)]. Then we show how the bounds on the global Tjurina number of \(V\) obtained by du Plessis and Wall lead to substantial improvements of our previous results on the stability of the reflexive sheaf \(T\langle V\rangle\) of logarithmic vector fields along \(V\), and on the Torelli property in the sense of I. Dolgachev and M. Kapranov [Duke Math. J. 71, No. 3, 633–664 (1993; Zbl 0804.14007)] of \(V\).
For the entire collection see [Zbl 1467.14002].
For the entire collection see [Zbl 1467.14002].
MSC:
| 14C34 | Torelli problem |
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
| 14B05 | Singularities in algebraic geometry |
| 14H20 | Singularities of curves, local rings |
| 32S05 | Local complex singularities |
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |
Keywords:
projective hypersurfaces; syzygies; logarithmic vector fields; stable reflexive sheaves; Torelli propertiesReferences:
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