Given an elliptically fibered Calabi-Yau threefold in Weierstrass form I want to compute the number of complex structure moduli of the fibration.

I know how it is done for the generic Weierstrass model: Let $\mathcal B$ be the base manifold. For definiteness let $\mathcal B = \mathbb C \mathbb P^2$. Its canonical bundle is $\bar K_B = \mathcal O (-3)$. Let $X: y^2 = x^3 + f xz^4 + g z^6$ with $f$ a section of $\bar K_B^{-4}$ and $g$ a section of $\bar K_B^{-6}$, i.e. on an operational level homogeneous degree 12 and 18 polynomials in 3 variables, respectively. The number of complex structure moduli is given by the number of ways one can deform $f$ and $g$ which is the number of coefficients. $f,g$ have $\binom{12+3-1}{12} = 91, \binom{18+3-1}{18} = 190$ coefficients. Polynomials which differ only by a $GL(3,\mathbb C)$ coordinate transformation are isomorphic. Therefore we have to subtract by $\mathrm{dim}_{\mathbb C} GL(3,\mathbb C) = 9$. All in all, we therefore have $91+190-9=272$ complex structure moduli.

Now, I am interested in the case where $f$ and $g$ are not general sections but where they are of a certain form, e.g. $f=f_3f_9+f_6^2$ where $f_i$ is a homogeneous polynomial of degree $i$.

Obviously, there should be less complex structure moduli but to me it is not obvious how to generalize the above calculations. Already a reference would be very helpful.


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