projectivelike

Projectivelike is an informal, nonstandard term used in mathematics to describe objects that resemble projective spaces in one or more essential respects. Because it is not a fixed technical definition, its precise meaning varies by author and context.

In practice, projectivelike is used to indicate that a space or family shares features typical of projective spaces. Common criteria include the existence of an ample line bundle that induces a projective embedding into some projective space via a complete linear system, and the presence of a graded coordinate-like algebra generated by global sections that mirrors the homogeneous coordinate ring of projective space. The idea can also be tied to symmetry or homogeneity properties reminiscent of projective spaces, such as those seen in certain homogeneous or flag varieties.

Examples where the notion may be invoked include actual projective spaces P^n, Grassmannians and flag varieties (which admit natural embeddings into projective spaces via Plücker or similar coordinates), and more generally smooth projective varieties equipped with ample line bundles that yield projective embeddings. Toric varieties with ample line bundles are another context where a projectivelike perspective is common. Objects that are not projective or lack a suitable embedding are typically not described as projectivelike.

Usage notes and caveats: because “projectivelike” is informal, authors should define precisely what properties they mean in a given work. The term is often used to discuss positivity, embeddings, and comparisons with projective space in topics such as birational geometry, moduli problems, and noncommutative or analytic analogues. See also projective space, ample line bundle, projective embedding, and homogeneous varieties.