I am interested in computing the (anti)-canonical class of the (total space of the) projective completion of the tautological bundle over $P^1\times P^1$. That is, the canonical class of $\mathbb P\_{P^1\times P^1}(J \oplus \mathscr O)$, where $J$ is the tautological line bundle on $P^1\times P^1$. 


I believe this can be done by computing the fan of the toric variety and summing the classes of the orbit closures of the one-skeleton? I was hoping for insight into perhaps a slicker/less cumbersome way of approaching this computation. 


Thanks in advance. 



How can one compute the canonical class of the projective completion of the tautological bundle over $P^1\times P^1$?
Why not use the Leray-Hirsch theorem? That says that the integral cohomology ring of a projectivized rank $n$ vector bundle $\pi: PE \to B$ is generated, as an algebra over the cohomology of the base $B$, by the first Chern class $h$ of the relative $O(1)$, with relation $h^n + c\_1 h^{n-1} + \dots + c\_n$, where $c\_i$ is the $i$th Chern class of $E$. 


On the other hand, you have a short exact sequence 
$$0 \longrightarrow T\_\pi \longrightarrow T\_{PE} \longrightarrow \pi^\* T\_B \longrightarrow 0,$$
where $T\_\pi$ is the relative tangent bundle. For a projectivized vector bundle, $T\_\pi = \mbox{Hom }(O(-1),E/O(-1))$ (this is a special case of the formula for the tangent bundle of a Grassmannian). Note that $\mbox{Hom }(O(-1),E/O(-1)) \cong E(1)/O$, so $\Lambda^{n-1} T\_\pi \cong (\Lambda^n E)(n)$ canonically.


Hence if $B$ is smooth of dimension $b$,
$$K\_{PE} = \Lambda^{b+n-1} T^\*\_{PE} \cong \Lambda^b \pi^\* T^\*\_B \otimes \Lambda^{n-1} T^\*\_\pi \cong \pi^\* K\_B \otimes \pi^\* (\Lambda^n E^\*)(-n).$$


Taking Chern classes, if $k\_{PE} := c\_1(K\_{PE})$ and $k\_b := c\_1(K\_B)$, we get
$$k\_{PE} = \pi^\* k\_B - \pi^\* c\_1(E) - nh.$$


As a check, note that if we replace $E$ by $E \otimes L$ for some line bundle $L$, then $PE$ is unchanged, but $c\_1(E)$ and $h$ are increased and decreased by $nc\_1(L)$ and $c\_1(L)$ respectively, so the formula above is unchanged.


In your case, the tautological line bundle $J \to P^1 \times P^1$ has $c\_1(J) = -h\_1-h\_2$ where $h\_i$ generate the cohomology of the two $P^1$ factors, so the canonical class, in terms of this basis, is $-2h\_1-2h\_2+h\_1+h\_2-2h = -h\_1-h\_2-2h$. Warning: as we saw, $h$ is NOT unchanged if we replace $E$ by $E \otimes L$! So if you replace $J \oplus O$ by, say, $O \oplus J^{-1}$, you will get an apparently different answer... 


