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.

# Logic Seminar

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Let k be a field of characteristic zero and K=k((t)). Semi-algebraic sets over K are boolean combinations of algebraic sets and sets defined by valuative inequalities. The associated Grothendieck ring has been studied by Hrushovski and Kazhdan who link it via motivic integration to the Grothendieck ring of varieties over k. I will present a morphism from the former to the Grothendieck ring of motives of rigid analytic varieties over K in the sense of Ayoub. This allows to refine the comparison by Ayoub, Ivorra and Sebag between motivic Milnor fibre and motivic nearby cycle functor.

It is well known that the theory of differentially closed fields of characteristic 0 has prime models and that they are unique up to isomorphism. One can ask the same question for the theory ACFA of existentially closed difference fields (recall that a difference field is a field with an automorphism).

In this talk, I will first give the trivial reasons of why this question cannot have a positive answer. It could however be the case that over certain difference fields prime models (of the theory ACFA) exist and are unique. Such a prime model would be called a difference closure of the difference field K. I will show by an example that the obvious conditions on K do not suffice.

I will then consider the class of aleph-epsilon saturated models of ACFA, or of kappa-saturated models of ACFA. There are natural notions of aleph-epsilon prime model and kappa-prime model. It turns out that for these stronger notions, if K is an algebraically closed difference field of characteristic 0, with fixed subfield F aleph-epsilon saturated, then there is an aleph-epsilon prime model over K, and it is unique up to K-isomorphism. A similar result holds for kappa-prime when kappa is a regular cardinal.

None of this extends to positive characteristic.

We discuss the connections and differences between the ZFC set theory and univalent foundations and answer the above question in the negative.