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This paper gives a geometric interpretation of bordered Heegaard Floerhomology for manifolds with torus boundary. If $M$ is such a manifold, we showthat the type D structure $\widehat{\mathit{CFD}}$ may be viewed as a set ofimmersed curves decorated with local systems in $\partial M$. Thesecurves-with-decoration are invariants of the underlying three-manifold up toregular homotopy of the curves and isomorphism of the local systems. Given twosuch manifolds and a homeomorphism $h$ between the boundary tori, the HeegaardFloer homology of the closed manifold obtained by gluing with $h$ is obtainedfrom the Lagrangian intersection Floer homology of the curve-sets. Thismachinery has several applications: We establish that the dimension of$\widehat{\mathit{HF}}$ decreases under a certain class of degree one maps(pinches) and we establish that the existence of an essential separating torusgives rise to a lower bound on the dimension of $\widehat{\mathit{HF}}$. Inparticular, it follows that a prime rational homology sphere $Y$ with$\widehat{\mathit{HF}}(Y)<5$ must be geometric. Other results include a newproof of Eftekhary's theorem that L-space homology spheres are atoroidal; acomplete characterisation of toroidal L-spaces in terms of gluing data; and aproof of a conjecture of Hom, Lidman, and Vafaee on satellite L-space knots.

Bordered Floer homology for manifolds with torus boundary via immersed curves