We introduce a new method to study mixed characteristic deformation of line bundles. In particular, for families <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$f : \mathcal {X} \rightarrow \mathcal {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi mathvariant="script">X</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">S</mi> </mrow> </math></EquationSource> </InlineEquation> with big monodromy and large period image defined over the ring of N-integers <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {O}_{L}[1/N]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">O</mi> <mi>L</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of a number field L, we produce a proper closed subscheme <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {E} \subsetneq \mathcal {S}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">E</mi> <mo>⊊</mo> <mi mathvariant="script">S</mi> </mrow> </math></EquationSource> </InlineEquation> outside of which all line bundles appearing in positive characteristic fibres of f admit characteristic zero lifts. This in particular applies to elliptic surfaces over <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {P}^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> and projective hypersurfaces in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {P}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> of degree <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$d \ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. We also study the locus <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {E}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">E</mi> </math></EquationSource> </InlineEquation> in more detail in the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$h^{0, 2} = 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>h</mi> <mrow> <mn>0</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> case.

Arithmetic deformation of line bundles

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2026年1月1日

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David Urbanik,
Ziquan Yang

摘要

We introduce a new method to study mixed characteristic deformation of line bundles. In particular, for families $f : \mathcal {X} \rightarrow \mathcal {S}$ $f : X \to S$ with big monodromy and large period image defined over the ring of N-integers $\mathcal {O}_{L}[1/N]$ $O_{L} [1 / N]$ of a number field L, we produce a proper closed subscheme $\mathcal {E} \subsetneq \mathcal {S}$ $E ⊊ S$ outside of which all line bundles appearing in positive characteristic fibres of f admit characteristic zero lifts. This in particular applies to elliptic surfaces over $\mathbb {P}^1$ $P^{1}$ and projective hypersurfaces in $\mathbb {P}^3$ $P^{3}$ of degree $d \ge 5$ $d \geq 5$ . We also study the locus $\mathcal {E}$ $E$ in more detail in the $h^{0, 2} = 2$ $h^{0, 2} = 2$ case.