Fields of Moduli of Smooth del Pezzo Surfaces

2026/07/17 by Tianzhi Yang
#math.AG

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Abstract

The field of moduli of a variety X over an algebraically closed field K is defined as the fixed field of those automorphisms σ of K for which X≃ X^σ. A fundamental question is under what conditions a variety admits a model over its field of moduli. We give a complete answer for smooth del Pezzo surfaces in characteristic 0: every smooth del Pezzo surface of degree at least 3 has a model over its field of moduli, whereas in degrees 1 and 2 there exist smooth complex del Pezzo surfaces with field of moduli ℝ which do not admit a real model.

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