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Contrary to previous approaches bringing together algebraic geometry andsignatures of paths, we introduce a Zariski topology on the space of pathsitself, and study path varieties consisting of all paths whose signaturesatisfies certain polynomial equations. Particular emphasis lies on the role ofthe non-associative halfshuffle, which makes it possible to describe varietiesof paths that satisfy certain relations all along their trajectory.Specifically, we may understand the set of paths on a given classical algebraicvariety in R d starting from a fixed point as a path variety. While halfshufflevarieties are stable under stopping paths at an earlier time, we furthermorestudy varieties that are stable under concantenation of paths. We point out howthe notion of dimension for path varieties crucially depends on the fact thatthey may be reducible into countably infinitely many subvarieties. Finally, wesee that studying halfshuffle varieties naturally leads to a generalization ofclassical algebraic curves, surfaces and affine varieties in finite dimensionalspace, where these generalized algebraic sets are now described throughiterated-integral equations. Keywords. path variety, shuffle ideal,halfshuffle, deconcatenation coproduct, tensor algebra, Zariski topology,concatenation of paths, Chen's identity, tree-like equivalence, regular map,generalized variety

An algebraic geometry of paths via the iterated-integrals signature