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

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.