Constructions of K-Regular Maps Using Finite Local Schemes

Explore the construction of k-regular maps using finite local schemes in this 58-minute lecture by Jarosław Buczyński. Delve into the historical problem of determining minimal dimensions for linearly independent point images, dating back to Chebyshev and Borsuk. Learn how algebraic geometry methods can be applied to construct k-regular maps and relate upper bounds to Hilbert scheme dimensions. Discover explicit examples for k ≤ 5 and upper bounds for arbitrary m and k. Examine the connection to interpolation theory and its implications for continuous functions on topological spaces. Follow the progression from manifolds and k-regular maps to secant varieties, Hilbert schemes, and scheme-theoretic methods for interpolation. Gain insights into various types of k-regularity and their applications in algebraic topology and geometry.

Syllabus

Intro
Manifolds
k-regular maps and interpolating subspaces
Setting
Non-monomial examples
Two constructions
Bounds for k-regular maps C
Bounds for any m, for k-regular maps C
The Ring
First k regular maps
Projections for embeddings
Projections and secant lines
Secant varieties
Projcetions for k-regular maps
Local approach
Local pictures
Punctual variant of a secant variety
Hilbert scheme
Why areoles?
Why Gorenstein?
Comparison
The return of the interpolation
Naive interpolation
Scheme theoretic method
Other types of k-regularity