I don't know if it belongs here but anyway, I need to compute arithmetic genus of divisors pulled back from a Fano base space to a bundle (which may or mayn't be trivial) defined through the projection map $\sigma: W\rightarrow P^3$, where $W$ is a $P^2$ bundle over $P^3$.
In this bundle W, construct a Calabi-Yau four-fold $X$ as a hypersurface and consider the divisor $D=σ^{−1}(C) ∩ X$, where $C$ is a divisor in the base $P^3$. I need to compute the $h^{i,0}$ (will write as $h^i$ below) of this divisor and consider the arithmetic genus of the divisor.
I have been told following is true for this divisor: $h^1=0, 0<h^2<h^3$ which seems to be derivable from Leray type spectral sequences given as proposition 2.2 in the paper "On minimal models of elliptic threefolds" by A Grassi which states that for an elliptic fibration between X (dimension n) and S with projection map $\pi: X\rightarrow S$ exists following exact sequence with $\Sigma$ a divisor of simple normal crossing (it means that D lies in the bundle C over $P^2$ here): $0\rightarrow H^l(S,\mathcal O_S)\rightarrow H^l(X,\mathcal O_X)\rightarrow H^{l-1}(S,\mathcal O_S(-\pi_*K_{X/S})\rightarrow 0$ with $0<l<n$.
This can be translated into usable forms such as below for (a 2 divisor in 3-space) eg. divisor W with a normal bundle N_W in the total space:
$H ^0 (D_3, O_{D3} ) = H ^0 (W_2, O_{W2} ), 0 → H ^1 (W_2, O_{W2} ) → H ^1 (D_{3}, O_{D3} ) → H^2(W_2, N_{W2} )^* → 0$, etc. but the problem is I don't know what the divisor here looks like or what it's normal bundle is and as a result it's not clear how to deduce $h^1=0, 0<h^2<h^3$ here? Or maybe there is another method?
Relevancy to physics: KKLT type ADS superpotentials canbe generated in this way and can be uplifted to ds metastable vacuum by anti-D branes to break Supersymmetry.
Edit: To see how it will work for a simple case. Here is an example: If the base is $P^3$, consider a point blown up in this base which gives a divisor isomorphic to $P^2$ with normal bundle $\mathcal O(-1)$ in the $P^3$ which gives following exact sequences: $$H ^0 (D, \mathcal O_{D})= H ^0 (W, \mathcal O_{W}),$$ $$0 → H ^1 (W, \mathcal O_{W} ) → H ^1 (D, \mathcal O_{D}) → H^2(W, N_{W})^* → 0,$$ $$0 → H ^2 (W, \mathcal O_{W} ) → H ^2 (D, \mathcal O_{D}) → H^1(W, N_{W})^* → 0,$$ $$H ^3 (D, \mathcal O_{D}) = H ^0 (W, N_{W})^*$$ where $(W, N_W)= (P^2, \mathcal O(-1))$ and D is the divisor in the total space about which information is retrieved with this spectral sequence giving $h^i=0$ for i=1,2,3. I need similar information in the above case with a divisor in $P^3$ given as a homogeneous polynomial of degree a>0. Thanks for any help.
