P1×P1

P1×P1 is the product of two projective lines over a field k. It is a smooth projective algebraic surface that can be realized as a nondegenerate quadric surface in P3 via the Segre embedding, mapping ([x0:x1], [y0:y1]) to [x0y0: x0y1: x1y0: x1y1], which satisfies x0x3 = x1x2.

Geometrically, P1×P1 has two natural projections to P1, giving two distinct rulings by copies of P1. The Picard group is free of rank two, generated by the classes of the two rulings, often denoted F1 and F2 or O(1,0) and O(0,1). A divisor of type (a,b) corresponds to a curve of bidegree (a,b) with respect to the two rulings. The intersection numbers satisfy F1^2 = F2^2 = 0 and F1·F2 = 1. The canonical class is K = -2F1 - 2F2, so the anticanonical class is -K = 2(F1+F2). The self-intersection K^2 = 8, making P1×P1 a del Pezzo surface of degree 8; it is also a minimal rational surface and the zeroth Hirzebruch surface, F0.

Automorphisms of P1×P1 form the group (PGL2 × PGL2) ⋊ Z/2, where the Z/2 factor exchanges the two rulings.

Line bundles and cohomology: for a,b ≥ 0, the space of global sections has dimension h0(O(a,b)) = (a+1)(b+1). The divisors O(a,b) are globally generated when a,b ≥ 0, and ample when a,b > 0.

Applications and role: P1×P1 serves as a standard example in algebraic geometry, illustrating ruled and rational surfaces, intersection theory, and the geometry of quadrics; it provides a concrete model for studying projections, morphisms, and moduli in a simple setting.