Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$X \subset \mathbb P^3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>⊂</mo> <msup> <mi mathvariant="double-struck">P</mi> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> be a nonsingular cubic hypersurface. Faenzi ([<CitationRef CitationID="CR8">8</CitationRef>]) and later Pons-Llopis and Tonini ([<CitationRef CitationID="CR21">21</CitationRef>]) have completely characterized ACM line bundles over X. As a natural continuation of their study in the non-ACM direction, in this paper, we completely classify <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-away ACM line bundles (introduced recently by Gawron and Genc ([<CitationRef CitationID="CR9">9</CitationRef>])) over X, when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\ell \le 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>≤</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\ell \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, we give examples of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-away ACM line bundles on X and for each <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\ell \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we establish the existence of smooth hypersurfaces <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$X^{(d)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>X</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> of degree <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$d >\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>></mo> <mi>ℓ</mi> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathbb P^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">P</mi> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> admitting <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-away ACM line bundles.

$\ell $-away ACM line bundles on a nonsingular cubic surface

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发表日期：

2026年6月24日

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Debojyoti Bhattacharya,
A. J. Parameswaran,
Jagadish Pine

摘要

Let $X \subset \mathbb P^3$ $X \subset P^{3}$ be a nonsingular cubic hypersurface. Faenzi ([8]) and later Pons-Llopis and Tonini ([21]) have completely characterized ACM line bundles over X. As a natural continuation of their study in the non-ACM direction, in this paper, we completely classify $\ell $ $ℓ$ -away ACM line bundles (introduced recently by Gawron and Genc ([9])) over X, when $\ell \le 2$ $ℓ \leq 2$ . For $\ell \ge 3$ $ℓ \geq 3$ , we give examples of $\ell $ $ℓ$ -away ACM line bundles on X and for each $\ell \ge 1$ $ℓ \geq 1$ , we establish the existence of smooth hypersurfaces $X^{(d)}$ $X^{(d)}$ of degree $d >\ell $ $d > ℓ$ in $\mathbb P^3$ $P^{3}$ admitting $\ell $ $ℓ$ -away ACM line bundles.

\(\ell \)-away ACM line bundles on a nonsingular cubic surface

摘要