Embedding Weil’s abstract varieties into projective space

Determine whether every abstract algebraic variety in the sense of André Weil’s Foundations of Algebraic Geometry (i.e., a variety defined by gluing local models in the Zariski topology) admits an embedding into projective space. Clarify necessary and sufficient conditions for such an embedding when it exists.

Background

Weil’s 1946 Foundations of Algebraic Geometry introduced “abstract varieties,” defined by patching local algebraic pieces analogously to manifolds, rather than as subsets of affine or projective space. This conceptual shift raised foundational questions about the relationship between abstractly defined varieties and classical projective embeddings.

In his 1950 ICM address, Zariski remarked that even basic representability issues about such abstract objects were unresolved. The question asks whether the abstract objects Weil defined can always be realized as projective varieties, i.e., embedded into some projective space.

References

An even more radical revision of the concept of a variety has been offered by André Weil. His so-called abstract varieties are not defined as subsets of the projective space, but are built out of pieces of ordinary varieties, pieces that must hang together in some well-defined fashion. It is still an open question whether the varieties of Weil can be embedded in the projective space.

— Constructing projective modules  (2412.05250 - Asok, 2024) in Entry “Foundations of algebraic geometry revisited” (A. Weil), Section A survey of pre-1951 mathematical culture