I have a CY four-fold as a hypersurface of degree $(4,3)$ in $P^3\times P^2$ and I have the projection map from this hypersurface say $X$ to $P^3$ as $\pi:X \rightarrow P^3$. Does this admit a section?
This has some application in String theory to certain instanton corrections.
No. By Lefschetz hyperplane theorem the Picard group of $X$ is generated by the pullbacks of the hyperplane classes of $\mathbb{P}^3$ and $\mathbb{P}^2$; this implies that the degree over $\mathbb{P}^3$ of every divisor on $X$ is divisible by 3. But if a projection to $\mathbb{P}^3$ has a section (even a rational section), it is a divisor of degree 1 over $\mathbb{P}^3$.