Bipartite algebraic graphs without quadrilaterals

We establish the result that if H∩(X×P2) is (2,2)-grid-free, then there exists F∈C[x¯,y¯] of degree ≤2 in y¯ such that H∩(X×P2)={F=0}∩(X×P2). Finally, we transfer the result to algebraically closed fields of large characteristic.