We show that for every toric surface X apart from <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {P}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {P}^1\times \mathbb {P}^1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>1</mn> </msup> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>1</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> and every ample line bundle <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation> on X there exists an ample polarisation A for X, such that the syzygy bundle <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M_{\mathcal {L}^{\otimes d}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <msup> <mrow> <mi mathvariant="script">L</mi> </mrow> <mrow> <mo>⊗</mo> <mi>d</mi> </mrow> </msup> </msub> </math></EquationSource> </InlineEquation> associated to the tensor power <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {L}^{\otimes d}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">L</mi> </mrow> <mrow> <mo>⊗</mo> <mi>d</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> is not stable with respect to A for every d sufficiently large.

On the instability of syzygy bundles on toric surfaces

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2025年9月6日

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Lucie Devey,
Milena Hering,
Katharina Jochemko,
Hendrik Süß

摘要

We show that for every toric surface X apart from $\mathbb {P}^2$ $P^{2}$ and $\mathbb {P}^1\times \mathbb {P}^1$ $P^{1} \times P^{1}$ and every ample line bundle $\mathcal {L}$ $L$ on X there exists an ample polarisation A for X, such that the syzygy bundle $M_{\mathcal {L}^{\otimes d}}$ $M_{L^{\otimes d}}$ associated to the tensor power $\mathcal {L}^{\otimes d}$ $L^{\otimes d}$ is not stable with respect to A for every d sufficiently large.