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

Daniele Agostini, Mario Kummer

TL;DR

This work fuses Ulrich sheaf theory with ${\mathbb A}^1$-enumerative geometry to define and compute arithmetically enriched invariants for space curves. It develops an apparatus where Ulrich (and symmetric Ulrich) sheaves induce relative orientations, enabling well-defined ${\mathbb A}^1$-degrees for finite projections and giving explicit Chow-form–level data via symmetric matrices. Central construction of an Ulrich sheaf on a secant variety yields an arithmetic writhe for curves in ${\mathbb P}^3$, an ${\mathbb A}^1$-analogue of Viro’s encomplexed writhe, invariant under algebraic isotopies. The paper further establishes algebraic isotopy theory for embeddings, delivering complete classifications for rational curves of low degree and revealing that higher-rank Ulrich sheaves produce new isotopy invariants, thereby enriching the landscape of arithmetic knot-like invariants in algebraic geometry.

Abstract

We establish a connection between the theory of Ulrich sheaves and $\mathbb{A}^1$-homotopy theory. For instance, we prove that the $\mathbb{A}^1$-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 $\mathbb{A}^1$-chain connected or $\mathbb{A}^1$-connected. Further if an embedded projective variety is the support of a symmetric Ulrich sheaf of rank one, the $\mathbb{A}^1$-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 $\mathbb{P}^3$. 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 $\mathbb{P}^3$ we obtain a complete classification up to algebraic isotopies.

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

TL;DR

This work fuses Ulrich sheaf theory with -enumerative geometry to define and compute arithmetically enriched invariants for space curves. It develops an apparatus where Ulrich (and symmetric Ulrich) sheaves induce relative orientations, enabling well-defined -degrees for finite projections and giving explicit Chow-form–level data via symmetric matrices. Central construction of an Ulrich sheaf on a secant variety yields an arithmetic writhe for curves in , an -analogue of Viro’s encomplexed writhe, invariant under algebraic isotopies. The paper further establishes algebraic isotopy theory for embeddings, delivering complete classifications for rational curves of low degree and revealing that higher-rank Ulrich sheaves produce new isotopy invariants, thereby enriching the landscape of arithmetic knot-like invariants in algebraic geometry.

Abstract

We establish a connection between the theory of Ulrich sheaves and -homotopy theory. For instance, we prove that the -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 -chain connected or -connected. Further if an embedded projective variety is the support of a symmetric Ulrich sheaf of rank one, the -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 . 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 we obtain a complete classification up to algebraic isotopies.
Paper Structure (19 sections, 37 theorems, 183 equations, 4 figures)

This paper contains 19 sections, 37 theorems, 183 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Ulrich sheaves
  4. Ulrich sheaves and the A1-degree
  5. Orientations and A1-degree
  6. Relative Ulrich sheaves and the A1-degree
  7. Symmetric Ulrich sheaves
  8. Secant varieties of curves
  9. Symmetric products of curves
  10. Line bundles on the symmetric product
  11. Secants and Viro frames
  12. Some cohomology
  13. Rational normal curves
  14. The arithmetic writhe
  15. An algebraic orientation on B-E
  16. ...and 4 more sections

Key Result

Lemma 4.16

Let $f\colon X\to Y$ be a finite surjective morphism of non-singular $K$-varieties, relatively oriented by the isomorphism Let $y\in Y$ a closed point outside the branch locus. The class of the fiber of $\tilde{\psi}$ at $y$ in $\operatorname{GW}(\kappa(y))$ is $\deg_y^{{\mathbb A}^1}(f)$.

Figures (4)

  • Figure 1: A knot diagram of the trefoil knot.
  • Figure 2: Local writhe numbers.
  • Figure 3: When applying a Reidemeister move of type I to a real algebraic curve, a crossing of two arcs becomes an isolated node.
  • Figure 4: The quartic curve considered in \ref{['ex:quarticwrith']} projected from the point $P$. The nodes of this planar curve correspond to secant lines containing $P$ and we denoted their local writhe in the picture.

Theorems & Definitions (120)

  • Definition : Arithmetic writhe
  • Definition 3.1: Ulrich sheaf
  • Example 3.2
  • Definition 4.1: Algebraic orientation
  • Example 4.2
  • Definition 4.3
  • Remark 4.4
  • Example 4.5
  • Definition 4.6: Relative orientation
  • Remark 4.7
  • ...and 110 more