Ulrich sheaves, the arithmetic writhe and algebraic isotopies of space curves

Agostini, Daniele; Kummer, Mario

doi:10.2140/gt.2026.30.1155

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Ulrich sheaves, the arithmetic writhe and algebraic isotopies of space curves

Daniele Agostini and Mario Kummer

Geometry & Topology 30 (2026) 1155–1201

DOI: 10.2140/gt.2026.30.1155

Abstract

We establish a connection between the theory of Ulrich sheaves and $𝔸^{1}$ -homotopy theory. For instance, we prove that the $𝔸^{1}$ -degree of a morphism between projective varieties, that is relatively oriented by an Ulrich sheaf, is constant on the target even when it is not $𝔸^{1}$ -chain connected or $𝔸^{1}$ -connected. Further if an embedded projective variety is the support of a symmetric Ulrich sheaf of rank one, the $𝔸^{1}$ -degree of all its linear projections can be read off in an explicit way from the free resolution of the Ulrich sheaf. Finally, we construct an Ulrich sheaf on the secant variety of a curve and use this to define an arithmetic version of Viro’s encomplexed writhe for curves in $ℙ^{3}$ . This can be considered to be an arithmetic analogue of a knot invariant. Namely, we define a notion of algebraic isotopy under which the arithmetic writhe is invariant. For rational curves of degree at most four in $ℙ^{3}$ we obtain a complete classification up to algebraic isotopies.

Keywords

Ulrich sheaf, secants of algebraic curves, arithmetic writhe, enriched knot theory

Mathematical Subject Classification

Primary: 14F42, 14H50, 14J60, 14N07

References

Publication

Received: 28 January 2025

Revised: 11 September 2025

Accepted: 10 October 2025

Published: 20 April 2026

Proposed: Marc Levine

Seconded: Kirsten Wickelgren, Arend Bayer

Authors

Daniele Agostini
Eberhard Karls Universität Tübingen Tübingen Germany

Mario Kummer
Technische Universität Dresden Germany

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Volume 30, issue 3 (2026)