[Paper Review] Surjective morphisms onto subelliptic varieties

Yuta Kusakabe|arXiv (Cornell University)|Dec 13, 2022

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper proves that every smooth subelliptic variety admits a surjective morphism from an affine space of dimension one more than the variety’s dimension, with the singular locus mapping to the whole variety. The proof uses algebro-geometric techniques involving sprays and pullbacks, generalizing results on flexible varieties and establishing jet interpolation for morphisms from zero-dimensional subschemes.

ABSTRACT

We prove that every smooth subelliptic variety admits a surjective morphism from an affine space. This result gives partial answers to the questions of Arzhantsev and Forstnerič. As an application, we characterize open images of morphisms between affine spaces. We also obtain the jet interpolation theorem for morphisms from zero-dimensional subschemes of affine varieties to smooth subelliptic varieties.

Motivation & Objective

• To resolve a question posed by Arzhantsev and Forstnerič on whether every smooth subelliptic variety admits a surjective morphism from an affine space.
• To generalize Arzhantsev’s result on very flexible varieties being images of affine spaces to the broader class of smooth subelliptic varieties.
• To establish a characterization of open images of morphisms between affine spaces in terms of codimension of the complement.
• To prove a jet interpolation theorem for morphisms from zero-dimensional subschemes to smooth subelliptic varieties.

Proposed method

• Utilizes the notion of sprays and dominating families of sprays to characterize subellipticity in algebraic geometry.
• Applies Serre’s Theorem A to construct surjective smooth morphisms from products of affine spaces to pullbacks of vector bundles.
• Employs the composed spray construction over the identity morphism of a subelliptic variety to generate dominant maps.
• Reduces the dimension of the source affine space from a large $\mathbb{A}^N$ to $\mathbb{A}^{\dim Y + 1}$ via affine embeddings of $d$-dimensional subspaces.
• Uses the quasi-compactness of $Y$ to cover the variety with finitely many images of affine subspaces under the morphism.
• Constructs a final morphism $f: \mathbb{A}^{\dim Y + 1} \to Y$ by composing with an embedding that maps distinct points to distinct subspaces covering $Y$.

Experimental results

Research questions

• RQ1 Does every smooth subelliptic variety admit a surjective morphism from an affine space, even without the properness assumption?
• RQ2 Can the open image of a morphism between affine spaces be characterized algebraically in terms of the codimension of the complement?
• RQ3 Does the jet interpolation property hold for morphisms from zero-dimensional subschemes to smooth subelliptic varieties?
• RQ4 Is there an algebro-geometric proof of the existence of surjective morphisms to subelliptic varieties, independent of analytic tools?
• RQ5 Can the dimension of the source affine space in such a surjection be bounded in terms of the target variety’s dimension?

Key findings

• Every smooth subelliptic variety $Y$ admits a surjective morphism $f: \mathbb{A}^{\dim Y + 1} \to Y$ such that $f(\mathbb{A}^{\dim Y + 1} \setminus \operatorname{Sing}(f)) = Y$.
• The result generalizes Arzhantsev’s theorem on very flexible varieties and answers his problem on sufficient conditions for being an image of an affine space.
• A Zariski open subset $U \subset \mathbb{A}^n$ is the image of a morphism from an affine space if and only if $\mathbb{A}^n \setminus U$ has codimension at least two.
• Every morphism from a zero-dimensional subscheme $Z \subset X$ to a smooth subelliptic variety $Y$ admits a lift to a morphism $\tilde{f}: X \to Y$.
• The proof is purely algebro-geometric and thus extends to smooth subelliptic varieties over arbitrary algebraically closed fields.
• The construction avoids analytic tools like the homotopy Runge approximation theorem, distinguishing it from Forstnerič’s original approach.