Let X be either a smooth K3 surface or a smooth Fano variety (i.e., <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$-K_X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi>K</mi> <mi>X</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is ample) of dimension n and index <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$i_X\ge n-2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>i</mi> <mi>X</mi> </msub> <mo>≥</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {E}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">E</mi> </math></EquationSource> </InlineEquation> be an initialized Ulrich bundle on X. In this paper, we show that the syzygy bundle <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$S_{\mathcal {E}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>S</mi> <mi mathvariant="script">E</mi> </msub> </math></EquationSource> </InlineEquation>, defined as the kernel of the evaluation map <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$eval:H^{0}(X,\mathcal {E})\otimes \mathcal {O}_{X}\rightarrow \mathcal {E},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>e</mi> <mi>v</mi> <mi>a</mi> <mi>l</mi> <mo>:</mo> <msup> <mi>H</mi> <mn>0</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi mathvariant="script">E</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊗</mo> <msub> <mi mathvariant="script">O</mi> <mi>X</mi> </msub> <mo stretchy="false">→</mo> <mi mathvariant="script">E</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> is semistable.

Stability of Syzygy Bundles of Ulrich Bundles

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2026年4月13日

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Rosa M. Miró-Roig

摘要

Let X be either a smooth K3 surface or a smooth Fano variety (i.e., $-K_X$ $- K_{X}$ is ample) of dimension n and index $i_X\ge n-2$ $i_{X} \geq n - 2$ and let $\mathcal {E}$ $E$ be an initialized Ulrich bundle on X. In this paper, we show that the syzygy bundle $S_{\mathcal {E}}$ $S_{E}$ , defined as the kernel of the evaluation map $eval:H^{0}(X,\mathcal {E})\otimes \mathcal {O}_{X}\rightarrow \mathcal {E},$ $e v a l : H^{0} (X, E) \otimes O_{X} \to E,$ is semistable.