Timeline for dualizing sheaf of deminormal variety
Current License: CC BY-SA 3.0
8 events
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| Sep 20, 2017 at 22:24 | comment | added | Kumar | @Sándor Kovács : yes this is what i wanted to make sure...i am in demi-normal case..i wanted to make sure that the dualizing sheaf can be represented by a Weil divisor which does not intersect the nodal locus at all...because if the Weil divisor D intersect the nodal locus then the ideal sheaf of this Weil divisor $\mathcal{O}(-D)$ will not be locally free at this intersection of D with this nodal locus...but the dualizing sheaf is locally free at all nodal points...so the Weil divisor which represents dualizing sheaf cannot intersect the nodal locus | |
| Sep 20, 2017 at 20:40 | comment | added | Sándor Kovács | I couldn't see the link in your last comment (google is random about this), but otherwise, the point is that the best you can do is what I wrote after "alternatively". In other words, Consider divisors that do not contain a codim $1$ comp of the sing locus. The Gor in codim $1$ assumption tells you that the canonical divisorial sheaf can be represented by such a divisor. However, this will not be true for a general divisorial sheaf. | |
| Sep 20, 2017 at 19:37 | comment | added | Kumar | also it is clearly written here (see the notation and conventions) books.google.co.in/… | |
| Sep 20, 2017 at 19:25 | comment | added | Kumar | : Please see the picture i have shared in the edited question ...this is from en.wikipedia.org/wiki/Canonical_bundle .......this says that there exist a Weil divisor class for dualizing sheaf if the variety is Gorenstein in codimension 1 and $S_2$. Also can you please elaborate "Alternatively, you can consider the associated divisors on the Gorenstein locus and insist that they never contain an irreducible component of the singular set.That might give you what you want."? | |
| Sep 20, 2017 at 19:09 | comment | added | Sándor Kovács | On non-normal schemes,you cannot consider divisors as cycles.If it is $S_2$, then the singular set itself is codimension $1$ and that messes up dealing with divisors the "usual" way.You should just stick to dealing with refelxive sheaves of rank $1$. If you really want "divisors" as opposed to a "linear system", then take sections, but the zero set of those sections will not behave exactly the same way.Alternatively, you can consider the associated divisors on the Gorenstein locus and insist that they never contain an irreducible component of the singular set.That might give you what you want. | |
| Sep 20, 2017 at 17:25 | comment | added | Kumar | How do we find this Weil divisor D(say), whose ideal sheaf is the dualizing sheaf of the deminormal variety X? I guess we have a Weil divisor D' on the smooth locus of X which gives the dualizing sheaf. Then we just take the closure of D'. Is it D? It is clear that if this is correct then D cannot intersect the codimension 1 nodal locus of the deminormal variety X..............the reference that you have shared is very helpful...is there some article which says how to find this Weil divisor? | |
| Sep 20, 2017 at 17:16 | vote | accept | Kumar | ||
| Sep 20, 2017 at 16:55 | history | answered | Sándor Kovács | CC BY-SA 3.0 |