Using the exact sequence

$$0\mapsto\mathcal{O}_{\mathbb{P}^{2}}\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(1)^{\oplus 3}\rightarrow T_{\mathbb{P}^{2}}\mapsto 0$$

it is easy to compute $H^{1}(\mathbb{P}^{2},T_{\mathbb{P}^{2}}) = H^{2}(\mathbb{P}^{2},T_{\mathbb{P}^{2}}) = 0$ while $h^{0}(\mathbb{P}^{2},T_{\mathbb{P}^{2}}) = 8$.

On the singular variety $\mathbb{P}(1,2,3)$ by $T_{\mathbb{P}(1,2,3)}$ I mean $\mathcal{H}om(\Omega_{\mathbb{P}(1,2,3)},\mathcal{O}_{\mathbb{P}(1,2,3)})$.

Is there an analogous way (or a completely different way) of computing the cohomology groups of $T_{\mathbb{P}(1,2,3)} = \mathcal{H}om(\Omega_{\mathbb{P}(1,2,3)},\mathcal{O}_{\mathbb{P}(1,2,3)})$ ?

If it helps $\mathbb{P}(1,2,3)$ can be embedded in $\mathbb{P}^{6}$ as a singular Del Pezzo surface of degree six.

up vote 8 down vote accepted

If you think about $P(1,2,3)$ as about stack then there is an analogue of the Euler sequence $$ 0 \to O \to O(1) \oplus O(2) \oplus O(3) \to T \to 0. $$ It allows to compute $h^1 = h^2 = 0$ and $h^0 = 5$.

  • 1
    Just to add one comment: away from small characteristic (2 and 3), the computation "should" be the same whether you work with $\mathbb{P}(1,2,3)$ as a stack or as a coarse moduli space. There is an action of $\mathbb{G}_m$ on $\mathbb{A}^3\setminus\{0\}$, and the affine, geometric quotient $q:(\mathbb{A}^3\setminus\{0\})\to \mathbb{P}(1,2,3)$ gives a way of realizing $\Theta$ as the invariant part of $q_* \widetilde{T}$, for $\widetilde{T}$ defined as Sasha suggests on $\mathbb{A}^3\setminus\{0\}$. Since $q$ is affine, compute for $\widetilde{T}$ and take $\mathbb{G}_m$-invariants. – Jason Starr Feb 11 '14 at 19:09

A sheaf D of differentials on any weighted projective space WPS have been constructed by I.Dolgachev (1982). He computed cohomology of the D(n)' s and generalized the Bott theorem to WPS' s. ( Before him, C.Delorme computed cohomology of the sheaves O(n) and studied duality for WPS' s (1975)).The ref. is : I.Dolgachev, Weighted projective varieties,in "Group Actions and Vector Fields" , Lect. N. Math. 956, Springer-Verlag, 1982,pp. 34-72. (ref. for C. Delorme is included).

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.