We study the arithmetic of del Pezzo surfaces Y of degree 2 over a function field, and in particular, the cokernel of the homomorphism from the Picard group to the Galois-invariants of the geometric Picard group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{Pic} \,Y \rightarrow (\textrm{Pic} \,\overline{Y})^{G}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Pic</mtext> <mspace width="0.166667em" /> <mi>Y</mi> <mo stretchy="false">→</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mtext>Pic</mtext> <mspace width="0.166667em" /> <mover> <mi>Y</mi> <mo>¯</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mi>G</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. Applying this to a fibration <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\pi :X\rightarrow S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation> in del Pezzo surfaces of degree 2 over a rational surface S, we construct examples with nontrivial relative unramified cohomology group <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^2_{nr,\pi }(k(X)/k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mrow> <mi>n</mi> <mi>r</mi> <mo>,</mo> <mi>π</mi> </mrow> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. A specialization argument implies the failure of stable rationality of varieties specializing to X.

Degree 2 del Pezzo surface bundles and stable rationality

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发表日期：

2026年1月14日

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Wenhao Li

摘要

We study the arithmetic of del Pezzo surfaces Y of degree 2 over a function field, and in particular, the cokernel of the homomorphism from the Picard group to the Galois-invariants of the geometric Picard group $\textrm{Pic} \,Y \rightarrow (\textrm{Pic} \,\overline{Y})^{G}$ $Pic Y \to {(Pic \bar{Y})}^{G}$ . Applying this to a fibration $\pi :X\rightarrow S$ $π : X \to S$ in del Pezzo surfaces of degree 2 over a rational surface S, we construct examples with nontrivial relative unramified cohomology group $H^2_{nr,\pi }(k(X)/k)$ $H_{n r, π}^{2} (k (X) / k)$ . A specialization argument implies the failure of stable rationality of varieties specializing to X.