Statements such as "there is a unique line between any pair of distinct points in the plane" and "there are 27 lines on any cubic surface" have given rise to the modern theory of enumerative geometry. To define such "curve counts" in a general setting usually involves choosing a particularly nice compactification of the space of smooth
embedded curves (one which admits a natural "virtual fundamental class").  I will propose a new perspective on enumerative invariants which is based instead on a certain "Grothendieck group of 1-cycles" and the "universal" curve enumeration invariant taking values in this group. It turns out that if we restrict to complex threefolds with
nef anticanonical bundle, this group has a very simple structure: it is generated by "local curves".This generation result implies some new cases of the MNOP conjecture relating Gromov--Witten and Donaldson--Pandharipande--Thomas invariants of complex threefolds.