[Paper Review] Bia{\l}ynicki-Birula decomposition for reductive groups

Joachim Jelisiejew, Łukasz Sienkiewicz|arXiv (Cornell University)|May 29, 2018

Algebraic Geometry and Number Theory22 references3 citations

TL;DR

This paper generalizes the classical Białynicki-Birula decomposition from $\mathbb{G}_m$-actions on smooth varieties to actions of arbitrary linearly reductive groups $\mathbf{G}$ on finite type schemes and algebraic spaces. It introduces a functorial construction using compactified monoids $\overline{\mathbf{G}}$ that parameterize $\mathbf{G}$-equivariant extensions of morphisms, proving representability by a scheme locally of finite type and showing that intersections of Gröbner cells in Hilbert schemes are isomorphic to affine spaces.

ABSTRACT

We generalize the Bia{\\l}ynicki-Birula decomposition from actions of $G_m$ on smooth varieties to actions of linearly reductive group ${\\bf G}$ on finite type schemes and algebraic spaces. We also provide a relative version and briefly discuss the case of algebraic stacks. We define the Bia{\\l}ynicki-Birula decomposition functorially: for a fixed ${\\bf G}$-scheme $X$ and a monoid $\\overline{\\bf G}$ which partially compactifies ${\\bf G}$, the BB decomposition parameterizes ${\\bf G}$-schemes over $X$ for which the ${\\bf G}$-action extends to the $\\overline{\\bf G}$-action. The freedom of choice of $\\overline{\\bf G}$ makes the theory richer than the $G_m$-case.

Motivation & Objective

• To extend the classical Białynicki-Birula decomposition from $\mathbb{G}_m$-actions to actions of arbitrary linearly reductive groups $\mathbf{G}$ on finite type schemes and algebraic spaces.
• To provide a functorial framework for the decomposition using partial compactifications $\overline{\mathbf{G}}$ of $\mathbf{G}$, generalizing the role of $\mathbb{A}^1$ in the $\mathbb{G}_m$-case.
• To establish representability of the Białynicki-Birula functor $\mathcal{D}_{X,\overline{\mathbf{G}}}$ by a scheme locally of finite type under mild assumptions, including existence of a zero in $\overline{\mathbf{G}}$.
• To analyze intersections of Gröbner cells in Hilbert schemes under different monomial orderings, showing they are isomorphic to affine spaces.

Proposed method

• Define the Białynicki-Birula functor $\mathcal{D}_{X,\overline{\mathbf{G}}}(S)$ as the set of $\mathbf{G}$-equivariant morphisms $\varphi: \overline{\mathbf{G}} \times S \to X$, where $\mathbf{G}$ acts on $\overline{\mathbf{G}}$ by left multiplication and trivially on $S$.
• Use the existence of a zero in $\overline{\mathbf{G}}$ and normality assumptions to reduce to a connected monoid with zero, ensuring representability.
• Prove that the functor $\mathcal{D}_{X,\overline{\mathbf{G}}}$ is representable by a scheme locally of finite type via formal and smoothness arguments.
• Apply the theory to the Hilbert scheme $\mathcal{H} = \mathrm{Hilb}_d(\mathbb{A}^2)$, using two $\mathbb{G}_m$-actions induced by different monomial orderings.
• Show that the fiber product $Z_1 \times_{\mathcal{H}} Z_2$ of two Białynicki-Birula cells corresponding to the same monomial ideal is isomorphic to an affine space via extension of equivariant morphisms from $\mathbb{A}^2 \setminus \{0\}$ to $\mathbb{A}^2$.
• Leverage smoothness of $\mathcal{H}$ and equivariant lifting to prove that families over $\mathbb{A}^2 \setminus \{0\}$ extend uniquely to $\mathbb{A}^2$, establishing canonical isomorphism between the fiber product and the cell in the $\mathbb{A}^2$-compactified decomposition.

Experimental results

Research questions

• RQ1 Can the classical Białynicki-Birula decomposition for $\mathbb{G}_m$-actions be generalized to actions of arbitrary linearly reductive groups $\mathbf{G}$ on finite type schemes and algebraic spaces?
• RQ2 What is the correct replacement for $\mathbb{A}^1$ in the functorial description of the decomposition when generalizing from $\mathbb{G}_m$ to $\mathbf{G}$?
• RQ3 Under what conditions is the Białynicki-Birula functor $\mathcal{D}_{X,\overline{\mathbf{G}}}$ representable by a scheme locally of finite type?
• RQ4 What is the geometric structure of the intersection of two Białynicki-Birula cells in the Hilbert scheme of points on $\mathbb{A}^2$ corresponding to different monomial orderings but the same initial ideal?
• RQ5 Is the intersection of finitely many Gröbner cells with the same initial ideal isomorphic to an affine space?

Key findings

• The Białynicki-Birula functor $\mathcal{D}_{X,\overline{\mathbf{G}}}$ is representable by a scheme locally of finite type for any finite type $\mathbf{G}$-scheme $X$, provided $\mathbf{G}$ is connected and $\overline{\mathbf{G}}$ has a zero.
• For perfect fields, every normal monoid $\overline{\mathbf{G}}$ with a zero is equivalent to a connected monoid $\overline{\mathbf{N}}$ with zero, so the assumption of connectedness and existence of zero is not restrictive.
• The intersection $Z_1 \times_{\mathcal{H}} Z_2$ of two Białynicki-Birula cells in the Hilbert scheme $\mathcal{H} = \mathrm{Hilb}_d(\mathbb{A}^2)$ corresponding to the same monomial ideal under different monomial orderings is isomorphic to an affine space.
• The dimension of this intersection is $\dim_k \mathrm{Hom}(M, S/M)_{\geq 0, \geq 0}$, the dimension of the non-negative graded part of the tangent space at the monomial ideal $[M]$.
• The same result extends to intersections of arbitrarily many such cells with the same initial ideal, all of which are isomorphic to affine spaces.
• The canonical isomorphism between the fiber product $Z_1 \times_{\mathcal{H}} Z_2$ and the cell in the $\mathbb{A}^2$-compactified decomposition arises from the unique extension of $\mathbb{G}_m \times \mathbb{G}_m$-equivariant morphisms from $\mathbb{A}^2 \setminus \{0\}$ to $\mathbb{A}^2$.