Higher rank Clifford's theorem on the smooth quadric

Brill-Noether theory of curves has played a crucial role in the study of curves and their moduli since the 19th century, and has been extensively studied by several authors. Clifford’s theorem provides a starting point in determining the emptiness of Brill-Noether loci by providing an upper bound on $h^0(L)$ for a line bundle $L$ on a smooth curve $C$ in terms of the degree of $L$. It also characterizes the cases for which equality holds. In this paper, we prove an analogous result for higher rank sheaves on ${\mathbb{P}}^1\times {\mathbb{P}}^1$. Depending on how nice the first Chern class is, and whether the sheaf has global generation properties, we prove sharp upper bounds on $h^0(E)$ for slope semistable sheaves $E$ in terms of $\mathrm{rk}(E)$ and $c_1(E)$. We also find that any $E$ achieving the bound is a twist of a Steiner-like bundle, or closely related to such a bundle. As part of our investigation, we show that general extensions of stable vector bundles on elliptic curves and del Pezzo surfaces are semistable.