Let X be a Fano threefold with index <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$i_X$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>i</mi> <mi>X</mi> </msub> </math></EquationSource> </InlineEquation> and fundamental line bundle <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathcal {O}}_X(h)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">O</mi> <mi>X</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We study <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>–semistable rank two bundles <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathcal {E}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">E</mi> </math></EquationSource> </InlineEquation> on X with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$c_1({\mathcal {E}})=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">E</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$h^0({\mathcal {E}}) \ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>h</mi> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">E</mi> <mo stretchy="false">)</mo> </mrow> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$h^1({\mathcal {E}}(-\lceil \frac{i_X}{2}\rceil h))=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>h</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">E</mi> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mrow> <mo>⌈</mo> <mfrac> <msub> <mi>i</mi> <mi>X</mi> </msub> <mn>2</mn> </mfrac> <mo>⌉</mo> </mrow> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.

A remark by Daniel Ferrand on bundles on Fano threefolds

期刊论文
发表日期：

2026年7月28日

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Gianfranco Casnati,
Angelo Felice Lopez

摘要

Let X be a Fano threefold with index $i_X$ $i_{X}$ and fundamental line bundle ${\mathcal {O}}_X(h)$ $O_{X} (h)$ . We study $\mu $ $μ$ –semistable rank two bundles ${\mathcal {E}}$ $E$ on X with $c_1({\mathcal {E}})=0$ $c_{1} (E) = 0$ , $h^0({\mathcal {E}}) \ne 0$ $h^{0} (E) \neq 0$ and $h^1({\mathcal {E}}(-\lceil \frac{i_X}{2}\rceil h))=0$ $h^{1} (E (- ⌈ \frac{i_{X}}{2} ⌉ h)) = 0$ .