Timeline for Gorenstein varieties: why the two definitions are equivalent?
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6 events
| when toggle format | what | by | license | comment | |
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| Nov 3, 2021 at 16:15 | comment | added | Misha Verbitsky | many thanks! I would try to find the reference by googling Kempf--Knudsen--Mumford--Saint-Donat and in Freitag and Kiehl. | |
| Nov 3, 2021 at 13:16 | comment | added | inkspot | Toric implies rational implies Cohen--Macaulay; this is in Kempf--Knudsen--Mumford--Saint-Donat. There do exist normal varieties that are Gorenstein in the CLS sense but not Cohen--Macaulay, such as cones over abelian varieties of dimension at least 2. For some other beautiful examples, see Freitag and Kiehl's paper on cusps of Hilbert modular varieties, Invent. Math. 24 (1974), 121–148. | |
| Nov 3, 2021 at 12:59 | comment | added | Misha Verbitsky | It is implicit in Gauntlett, Martelli, Sparks, Yau that the singularity is rational (they consider the cone of the anticanonical embedding of a smooth Fano manifold), and I suppose that in Cox-Little-Schenk it is toric, so maybe they factor this in, and silently assume the Cohen-Macaulay property. I am not sure if Cohen-Macaulay follows from toric or from rational, though. | |
| Nov 3, 2021 at 12:54 | comment | added | Misha Verbitsky | However, Cox-Little-Schenk and Gauntlett, Martelli, Sparks, Yau nowhere assume that the singularity is Cohen-Macaulay. I thought it should follow from the rest of the assumptions, but in afterthought I am no longer sure it follows. | |
| Nov 3, 2021 at 12:47 | comment | added | Misha Verbitsky | Many thanks! I just need a reference to the most restrictive form of the statement, with isolated, normal, canonical singularity of Calabi-Yau type. However, the definition in Cox-Little-Schenk already assumes normality, so (even if there are non-normal examples) there are no contradictions. | |
| Nov 3, 2021 at 12:20 | history | answered | Donu Arapura | CC BY-SA 4.0 |