An asymptotic Nullstellensatz for curves
Abstract
Hilbert's Nullstellensatz asserts the existence of a complex point satisfying lying on a given variety, provided there is no (ideal-theoretic) proof to the contrary.
I will describe an analogue for curves (of unbounded degree), with respect to conditions specifying that they lie on a given smooth variety, and have homology class
near a specified ray. In particular, an analogue of the Lefschetz principle (relating large positive characteristic to characteristic zero) becomes available for such questions.
The proof is very close to a theorem of Boucksom-Demailly-Pau-Peternell on moveable curves, but requires a certain sharpening. This is part of a joint project with Itai Ben Yaacov, investigating the logic of the product formula; the algebro-geometric statement is needed for proving the existential closure of $\Cc(t)^{alg}$ in this language.
