The Maximal Free Rational Quotient

This short, expository note proves existence of the maximal quotient of a variety by free rational curves. 1. Definition of a maximal free rational quotient Definition 1.1. Let V be a Deligne-Mumford stack over a field k, and denote the smooth locus by V sm ⊂ V . A 1-morphism f : Pk → V sm is a free rational curve to V if fTV is generated by global sections and has positive degree. Let S be an irreducible algebraic space, let πX : X → S be a proper, locally finitely presented 1-morphism of Deligne-Mumford stacks with integral geometric generic fiber, and let X ⊂ X be a normal dense open substack. Denote by πX : X → S the restriction of πX . Definition 1.2. A free rational quotient of πX is a triple (X , Q, φ) whereX ⊂ X is a dense open substack, where Q is a normal algebraic space, finitely presented over S with integral geometric generic fiber, and where φ : X → Q is a dominant 1-morphism of S-stacks satisfying, (i) the geometric generic fiber F of φ is integral, and (ii) a general pair of distinct points of F is contained in the image of a free rational curve. A free rational quotient is trivial if φ : X → Q is birational, and nontrivial otherwise. A free rational quotient (X, Q, φ : X → Q) is maximal if for every free rational quotient (X 1 , Q ∗ 1, φ1 : X ∗ 1 → Q ∗ 1) there exists a dense open subset U ⊂ Q ∗ 1 and a smooth morphism ψ : U → Q such that φ|φ−1 1 (U) = ψ ◦ φ1. Theorem 1.3. There exists a maximal free rational quotient. It is not true that a maximal free rational quotient is unique, but it is unique up to unique birational equivalence of Q. 2. Proof of Theorem 1.3 The proof is very similar to the proofs of existence of the rational quotient in [1] and [2]. Existence of the maximal free rational quotient can be deduced from theorems there. However there are 2 special features of this case: The relation of containment in a free rational curve is already a rational equivalence relation, so existence of the quotient is less technical than the general case. Also, unlike the Date: February 2, 2008.