<p>We study the deformation theory of the Fano scheme <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{F}=\textrm{F}(\textrm{X})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>F</mtext> <mo>=</mo> <mtext>F</mtext> <mo stretchy="false">(</mo> <mtext>X</mtext> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of lines on a cubic <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textrm{X}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>X</mtext> </math></EquationSource> </InlineEquation> of dimension <i>d</i> with only finitely many singularities. By taking the relative Fano scheme, we define a morphism <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\eta :\mathcal {D}_{\textrm{X}}\rightarrow \mathcal {D}_{\textrm{F}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>η</mi> <mo>:</mo> <msub> <mi mathvariant="script">D</mi> <mtext>X</mtext> </msub> <mo stretchy="false">→</mo> <msub> <mi mathvariant="script">D</mi> <mtext>F</mtext> </msub> </mrow> </math></EquationSource> </InlineEquation> of the local moduli functors associated to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textrm{X}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>X</mtext> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{F}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>F</mtext> </math></EquationSource> </InlineEquation>, respectively. We show that for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$d\geqslant 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>⩾</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation> yields an isomorphism on first-order deformations; in particular, <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation> is an isomorphism whenever <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textrm{H}^{0}(\Theta _{\textrm{X}})=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mtext>H</mtext> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Θ</mi> <mtext>X</mtext> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Deformations of the Fano scheme of a cubic

期刊论文
发表日期：

2025年10月8日

查看全文

Samuel Stark

摘要

We study the deformation theory of the Fano scheme $\textrm{F}=\textrm{F}(\textrm{X})$ $F = F (X)$ of lines on a cubic $\textrm{X}$ $X$ of dimension d with only finitely many singularities. By taking the relative Fano scheme, we define a morphism $\eta :\mathcal {D}_{\textrm{X}}\rightarrow \mathcal {D}_{\textrm{F}}$ $η : D_{X} \to D_{F}$ of the local moduli functors associated to $\textrm{X}$ $X$ and $\textrm{F}$ $F$ , respectively. We show that for $d\geqslant 5$ $d ⩾ 5$ , $\eta $ $η$ yields an isomorphism on first-order deformations; in particular, $\eta $ $η$ is an isomorphism whenever $\textrm{H}^{0}(\Theta _{\textrm{X}})=0$ $H^{0} (Θ_{X}) = 0$ .