Artin-Tate Conjecture, fibered surfaces, and minimal regular proper model

Let K be a global function field of characteristic p, with finite field of constants k (the algebraic closure of Fp in K) of size q, so K is the function field k(V ) of a unique smooth, proper, geometrically connected curve V over k. Let XK be a smooth proper geometrically connected curve over K, with genus g > 0. The Jacobian J = Pic0XK/K ofXK is an abelian variety overK with dimension g > 0. The goal is to interpret the BSD conjecture for J in terms of the geometry of a smooth projective surface X over k equipped with a proper flat map X → V having generic fiber XK as given. The key point is to express the conjecture in terms of such an X without reference to the fibration. That such an X exists at all is already a real theorem, so we begin by discussing (without proofs) the story of “nice” proper flat models for curves.