Curves on a quadric surface over $overline{mathbb {F}}_p$

Here (following a paper by Giuffida, Maggioni and Re) we study the existence of curve $Cin vert mathcal {O}_{mathbb {P}^1 imes mathbb {P}^1}(a,b)vert$ defined over small fields and with $h^0(C,mathcal {O}_C(x,y))cdot h^1(C, mathcal {O}_C(x,y))=0$ for all integers $x, y$ such that $xge a$ and $yle -2$.