[Paper Review] Rationality does not specialize among terminal varieties

Burt Totaro|arXiv (Cornell University)|Aug 5, 2015

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper demonstrates that rationality does not specialize in families of terminal projective varieties, even in dimension 5 and above. Using a family of quartic 5-folds with terminal singularities, it shows that general fibers are rational while the central fiber (a cone over a non-stably rational 4-fold) is not, proving rationality fails to specialize among terminal varieties.

ABSTRACT

A limit of rational varieties need not be rational, even if all varieties in the family are projective and have at most terminal singularities.

Motivation & Objective

• To investigate whether rationality specializes in families of projective varieties with mild singularities, particularly terminal singularities.
• To resolve the open question of whether rationality specializes among terminal 4-folds or smooth varieties.
• To extend previous results on non-specialization of rationality in higher-dimensional varieties with canonical or klt singularities.
• To construct explicit examples in dimensions ≥5 where rationality fails to specialize despite terminal singularities and rational general fibers.

Proposed method

• Construct a flat family of quartic 5-folds in P^6 over a Zariski open subset of the complex affine line.
• Define the family via a pencil of hypersurfaces: f_4(x_0,…,x_5) + a·g_3(x_0,…,x_5)·x_6 = 0, where f_4 is a quartic and g_3 a cubic form.
• Show that for a ≠ 0, each fiber has multiplicity 3 at [0,…,0,1], satisfying the condition for rationality via Lemma 1.2.
• Prove that the central fiber (a = 0) is the projective cone over a non-stably rational 4-fold, hence not rational.
• Establish that terminal singularities are Zariski-open in families, so the general fibers are terminal.
• Extend the construction to 4-folds with canonical singularities by replacing the 4-fold with a non-stably rational 3-fold.

Experimental results

Research questions

• RQ1 Does rationality specialize among terminal varieties of dimension 5 or higher?
• RQ2 Does rationality specialize among terminal 4-folds or smooth varieties?
• RQ3 Can non-specialization of rationality be demonstrated in families with terminal singularities?
• RQ4 Is there a family of terminal varieties where general fibers are rational but the central fiber is not?
• RQ5 Can the non-specialization of rationality be extended to higher dimensions using cone constructions?

Key findings

• Rationality does not specialize among terminal varieties of dimension at least 5, as shown by a family of quartic 5-folds with rational general fibers and a non-rational terminal central fiber.
• The central fiber is the projective cone over a non-stably rational 4-fold, which is not rational because it is birational to P^1 × Y, where Y has a nonzero holomorphic 1-form.
• For a ≠ 0, each fiber in the family is rational due to multiplicity 3 at a rational point, satisfying the criterion in Lemma 1.2.
• Terminal singularities are preserved in a Zariski open neighborhood of the base, ensuring all fibers over C − {0} are terminal.
• The construction extends to 4-folds with canonical singularities by using a non-stably rational 3-fold as the base, showing non-specialization in dimension 4.
• The example generalizes to all dimensions ≥5 by taking products with P^m, preserving non-rationality of the central fiber.