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edited Oct 23 at 8:38

Vanising Vanishing of Ext group

Let $C$ be a cartier divisor on a smooth projective surface in $\mathbb{P}^3$. We then Then we get the short exact sequence ,$0 $0 \to \mathcal{O}_X(-C) \to \mathcal{O}_X(-C_{red}) \to F \to 0$ 0$$ for some sheaf $F$. As we We see that $F$ is supported on $C$. Assuming $C \not= C_{red}$ when is it possible to say that $Ext^2_X(F,\mathcal{O}_X)=0$?

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asked Oct 23 at 8:33

Naga Venkata
556●2●6

Vanising of Ext group

Let $C$ be a cartier divisor on a smooth projective surface in $\mathbb{P}^3$. We then get the short exact sequence,

       $0 \to \mathcal{O}_X(-C) \to \mathcal{O}_X(-C_{red}) \to F \to 0$

for some sheaf $F$. As we see that $F$ is supported on $C$. Assuming $C \not= C_{red}$ when is it possible to say that $Ext^2_X(F,\mathcal{O}_X)=0$?