Nef cone and pseudoeffective cone are fundamental invariants of a smooth projective complex variety and an active area of research in Algebraic Geometry. Let $X=\mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$ , where $C$ is any smooth irreducible curve and $E_1$, $E_2$ are vector bundles over $C$. In this talk, we will discuss about the nef cone and pseudoeffective cone of $X$ whenever $E_1$ and $E_2$ are semistable, and will show that they coincide. We will also discuss about the nef cone and pseudoeffective cone of $X$, in case, both $E_1$ and $E_2$ have rank 2, and neither $E_1$ nor $E_2$ is semistable or at least one of them is semistable. Consequently, in the case, when both $E_1$ and $E_2$ are of rank 2 bundles on $C$, we show that nef cone and pseudoeffective cone of $X$ coincide iff both $E_1$ and $E_2$ are semistable.
This is a joint work with Rupam Karmakar and Nabanita Ray and has appeared in Bull. Sci. Math, 151,(2019) 1-12.
