[Paper Review] The maximal free rational quotient

Jason Starr|ArXiv.org|Feb 28, 2006

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper establishes the existence of a maximal free rational quotient for a variety over a field, defined as a dominant morphism to a normal algebraic space that parametrizes equivalence classes of free rational curves under rational equivalence. The construction uses moduli spaces of stable maps and Hilbert schemes, with the key result being that such a quotient exists and is unique up to birational equivalence of the base space.

ABSTRACT

This short, expository note proves the existence of the maximal quotient of a variety by free rational curves.

Motivation & Objective

• To define and construct the maximal free rational quotient of a variety under the action of free rational curves.
• To establish existence of such a quotient in the context of Deligne-Mumford stacks with proper, finitely presented morphisms.
• To show that the maximal free rational quotient is unique up to unique birational equivalence of the base space.
• To demonstrate that the quotient construction avoids the need for purely inseparable base changes, unlike general rational quotients.

Proposed method

• Define a free rational curve as a morphism from P¹ to the smooth locus of a stack with globally generated tangent pullback and positive degree.
• Construct the moduli stack H of free rational curves in fibers of the smooth locus of πX.
• Use the universal family of maps to define a correspondence W_i ⊂ X ×_S X as the closure of the image of (H_i × P¹ × P¹) under evaluation.
• Apply the Hilbert scheme functor to parametrize flat families of subschemes, and use Stein factorization to construct the quotient morphism φ: X* → Q*.
• Prove that the geometric generic fiber of φ is integral and that any general pair of points in a fiber is connected by a free rational curve.
• Use faithfully flat descent and properties of Hilbert schemes to show that the quotient satisfies the maximality condition.

Experimental results

Research questions

• RQ1 Does there exist a maximal quotient of a variety by free rational curves, satisfying a universal property under dominant morphisms?
• RQ2 Can such a maximal quotient be constructed without base change in positive characteristic, unlike general rational quotients?
• RQ3 Is the maximal free rational quotient unique, and if not, in what sense is it unique?
• RQ4 How does the geometric structure of the quotient relate to the existence of free rational curves in fibers?

Key findings

• A maximal free rational quotient (X*, Q*, φ) exists for any proper, locally finitely presented morphism πX: X → S with integral geometric generic fiber.
• The maximal free rational quotient is unique up to unique birational equivalence of the base space Q*.
• The construction does not require purely inseparable base change, unlike the general rational quotient construction.
• The quotient morphism φ: X* → Q* satisfies the condition that any general pair of points in a geometric fiber is connected by a free rational curve.
• The quotient is constructed via the Hilbert scheme of the total space and Stein factorization, ensuring Q* is normal and finitely presented over S.
• The maximal free rational quotient is trivial if and only if no free rational curves exist in the geometric generic fiber.