On flexibility of affine factorial varieties

1. Problem Statement

The paper addresses two central problems in algebraic geometry:

1. Factoriality of Suspensions: Determining precise conditions under which a "suspension" of an affine variety remains a factorial variety (a Unique Factorization Domain, or UFD).
2. Homogeneous Varieties vs. Homogeneous Spaces: Constructing examples of flexible affine varieties that are factorial but are not homogeneous spaces of algebraic groups.

While it is known that homogeneous spaces of algebraic groups are often flexible and factorial, the converse is not true. The authors aim to expand the class of known flexible factorial varieties using the suspension construction and distinguish them from classical homogeneous spaces using topological and algebraic invariants.

2. Methodology

The authors employ a combination of commutative algebra, geometric invariant theory, and algebraic topology:

• Suspension Construction: They utilize the suspension $X = \text{Susp}(Y, f)$, defined as the hypersurface $uv = f$ in $\mathbb{A}^2 \times Y$, where $Y$ is an affine variety and $f \in K[Y]$ is a non-constant regular function.
• Commutative Algebra: The core of the factoriality proof involves analyzing the ring $S = R[u, v]/(uv - f)$, where $R = K[Y]$. The authors use $\mathbb{Z}$-gradings, properties of prime and irreducible elements, and integral closure to establish criteria for $S$ being a UFD.
• Flexibility Theory: They rely on the concept of flexible varieties, where the special automorphism group $\text{SAut}(X)$ (generated by additive group actions $\mathbb{G}_a$) acts infinitely transitively on the smooth locus.
• Topological Invariants: To distinguish varieties, they compute homotopy groups ($\pi_k$) and homology groups ($H_k$), specifically utilizing the relationship between the suspension and the base hypersurface $\{f=0\}$.
• Divisor Class Groups: They compute the divisor class group $\text{Cl}(X)$ to provide an alternative geometric proof of the factoriality criterion.

3. Key Contributions and Results

A. Criterion for Factoriality (Theorem 1)

The paper establishes a necessary and sufficient condition for the suspension to be factorial.

• Result: Let $R$ be a domain and $f \in R$ be a non-zero, non-invertible element. The ring $S = R[u, v]/(uv - f)$ is a Unique Factorization Domain (UFD) if and only if:
  1. $R$ is a UFD.
  2. $f$ is a prime element in $R$.
• Significance: This generalizes the known result for Laurent polynomial rings ($R[T, T^{-1}]$) to the context of suspensions. The proof involves showing that if $f$ is prime, the elements $u$ and $v$ in $S$ are prime, and that the ascending chain condition on principal ideals is preserved.

B. Smoothness and Flexibility (Theorem 2)

Combining the factoriality criterion with previous results on smoothness, the authors derive a construction method for specific types of varieties.

• Result: If $Y$ is a flexible, smooth, affine, factorial variety, then the suspension $X = \text{Susp}(Y, f)$ is a smooth, flexible, affine, factorial variety if and only if:
  1. $f$ is a prime element in $K[Y]$.
  2. The hypersurface $\{f = 0\}$ is a smooth subvariety of $Y$.
• Significance: This provides an iterative procedure to generate infinite families of flexible factorial varieties in any dimension $\ge 3$.

C. Topological Properties (Proposition 3)

The authors analyze the topology of suspensions over $\mathbb{C}$.

• Result: If $Y$ is a simply connected, irreducible, normal affine variety, then the suspension $X = \text{Susp}(Y, f)$ is also simply connected.
• Significance: This allows the authors to rule out certain isomorphisms with known homogeneous spaces based on fundamental groups.

D. Main Counterexample (Theorem 3)

The paper constructs a specific counterexample to the question of whether all flexible factorial varieties are homogeneous spaces.

• Construction: Consider the hypersurface $X \subset \mathbb{C}^4$ defined by:
  $$x^2y - xy - uv + 1 = 0$$
  This is the suspension $X = \text{Susp}(\mathbb{C}^2, (x-1)xy + 1)$.
• Properties:
  • $X$ is a smooth, flexible, affine, factorial 3-fold.
  • $X$ is simply connected ($\pi_1(X) = 0$).
  • The third homotopy group is $\pi_3(X) \cong \mathbb{Z}_2$.
• Comparison: The only known homogeneous spaces of algebraic groups that are affine factorial 3-folds without non-constant invertible functions are $\mathbb{C}^3$, $\text{SL}(2)$, and $\text{SL}(2)/2I$ (where $2I$ is the binary icosahedral group).
  • $X \not\cong \mathbb{C}^3$ because $\pi_3(\mathbb{C}^3) = 0$ while $\pi_3(X) = \mathbb{Z}_2$.
  • $X \not\cong \text{SL}(2)$ because $\pi_3(\text{SL}(2)) \cong \mathbb{Z}$.
  • $X \not\cong \text{SL}(2)/2I$ because $X$ is simply connected, whereas $\text{SL}(2)/2I$ has a non-trivial fundamental group.
• Conclusion: $X$ is a homogeneous affine factorial variety that is not a homogeneous space of an algebraic group.

E. Divisor Class Group (Section 4)

The authors compute the divisor class group $\text{Cl}(X)$ for a suspension.

• Result: If $Y$ is factorial and $f = p_1^{a_1} \dots p_s^{a_s}$ is the prime factorization of $f$, then:
  $$\text{Cl}(X) \cong \mathbb{Z}^s / \langle (a_1, \dots, a_s) \rangle$$
• Significance: This confirms that $X$ is factorial ($\text{Cl}(X)=0$) if and only if $s=1$ and $a_1=1$ (i.e., $f$ is prime), offering a geometric proof of Theorem 1.

F. Limitations of Characterization (Section 5)

The authors show that the fiber structure of the $\mathbb{G}_m$-action (where generic fibers are hyperbolas and special fibers are unions of lines) is not sufficient to characterize a variety as a suspension. They provide an example of a 3-fold with this fiber structure that is not a suspension because the associated graded module is not cyclic.

4. Significance

• Expansion of Known Classes: The paper significantly expands the known class of flexible factorial varieties, moving beyond the realm of homogeneous spaces of algebraic groups.
• Topological Distinction: It demonstrates that topological invariants (specifically higher homotopy groups like $\pi_3$) are powerful tools for distinguishing between flexible varieties that share the same dimension and algebraic properties (factoriality).
• Algebraic Geometry: The rigorous criterion for the factoriality of suspensions ($R$ is UFD + $f$ is prime) provides a robust tool for constructing new examples of UFDs in algebraic geometry.
• Counterexamples: The construction of the specific 3-fold $X$ resolves a subtle question regarding the classification of flexible varieties, proving that the class of flexible varieties is strictly wider than the class of homogeneous spaces of algebraic groups, even within the factorial category.