Severi–Brauer Varieties in Algebraic Geometry

• Severi–Brauer varieties are smooth, projective varieties defined over a field that become isomorphic to projective space after base change, linking them to central simple algebras.
• They encapsulate twisted forms of projective spaces and provide a geometric realization of Brauer group elements, with key invariants like period and index.
• Their universal construction aids in addressing period–index problems and discriminant avoidance, underpinning advances in both arithmetic and noncommutative algebraic geometry.

A Severi–Brauer variety is a smooth, projective variety over a field $k$ that becomes isomorphic to projective space after base change to an algebraic closure $\bar{k}$. The theory of Severi–Brauer varieties fundamentally connects central simple algebras, the Brauer group, and twisted forms of projective varieties, forming a core segment of the study of noncommutative algebraic geometry and arithmetic geometry. The modern perspective also includes the construction of universal classifying spaces for these objects, facilitating deeper analysis of period–index phenomena and applications in both arithmetic and geometry.

1. Foundational Definitions: Central Simple Algebras and Severi–Brauer Varieties

Let $k$ be a field. A finite-dimensional $k$-algebra $A$ is central simple if it has center exactly $k$ and no nontrivial two-sided ideals. Given such an $A$ of degree $n$ ($\dim_k A = n^2$), the associated Severi–Brauer variety $\mathrm{SB}(A)$ parametrizes right ideals $I\subset A$ with $\dim_k I = n$; it is a smooth projective $k$-variety of dimension $n-1$. After base change to $\bar{k}$, one has $$\mathrm{SB}(A)_{\bar{k}} \cong \mathbb{P}^{n-1}_{\bar{k}}$$, making $\mathrm{SB}(A)$ a twisted form of projective space. Conversely, every such twisted form arises in this way from a unique (up to isomorphism) central simple algebra, setting up a bijection between isomorphism classes of Severi–Brauer varieties of dimension $n-1$ and elements of $\mathrm{Br}(k)[n]$, the n-torsion of the Brauer group $[2510.20474]$.

The class $[A]$ in $\mathrm{Br}(k)$ corresponds to the cohomological obstruction to the triviality of the Severi–Brauer variety.

2. Period, Index, and the Brauer Group Structure

Given a Severi–Brauer variety $P = \mathrm{SB}(A)$ of degree $n$, its class $\alpha \in \mathrm{Br}(k)$ possesses two numerical invariants:

• Period ($\mathrm{per}(\alpha)$): The order of $\alpha$ in the torsion group $\mathrm{Br}(k)$.
• Index ($\mathrm{ind}(\alpha)$): The degree $m$ of the unique central division algebra Brauer-equivalent to $A$.

These invariants satisfy $$\mathrm{per}(\alpha)\mid \mathrm{ind}(\alpha) \mid \mathrm{per}(\alpha)^{\dim(k)-1}$$, and for a Severi–Brauer variety $P$ corresponding to a class of period $d$, the minimal multiple $d$ such that $\mathcal{O}_P(d)$ descends to $P$ from $\mathbb{P}^{n-1}_{\bar{k}}$ is precisely the period. The index is the minimal $m$ such that $P$ admits a $k$-rational linear subspace of dimension $m-1$ $[2510.20474]$.

Via the dictionary

$\{\text{Central Simple Algebras of degree %%%%0%%%%}\}/\cong \longleftrightarrow \{\text{Severi–Brauer varieties of dim %%%%1%%%%}\}/\cong \longleftrightarrow \mathrm{Br}(k)[n],$

one often encodes the equivalence of nontrivial forms of projective space with nontrivial Brauer classes.

3. Universal Severi–Brauer Varieties and Classifying Space Construction

For fixed integers $n, d$ with $d \mid n$, Gounelas and Huybrechts construct a smooth quasi-projective variety $B_{n,d}$ together with a universal Severi–Brauer family $\pi: Q \rightarrow B$, with these properties $[2510.20474]$:

• $Q \rightarrow B$ is a Severi–Brauer variety of dimension $n-1$, with $\mathrm{per}([Q]) = d$, $\mathrm{ind}([Q]) = n$, and $[Q]$ generating $\mathrm{Br}(B) \cong \mathbb{Z}/d\mathbb{Z}$.
• For any Severi–Brauer variety $P \rightarrow X$ over a quasi-projective base (of dimension up to $m$) with period dividing $d$, there exists a classifying map $\phi: X \to B$ such that $P \cong \phi^* Q$.

The construction exploits the $d$-th Veronese embedding, the natural Hilbert scheme parameterizing embeddings of $\mathbb{P}^{n-1}$ via $\mathcal{O}(d)$, and geometric invariant theory: $B$ is realized as a GIT orbit in the Hilbert scheme of $\mathbb{P}^{N-1}$, where $N = \binom{n+d-1}{n-1}$.

Table: Key Structural Invariants of $B_{n,d}$

Case	$\pi_1(B)$	$\mathrm{Pic}(B)$	$\mathrm{Br}(B)$
$d < n$	0	$\mathbb{Z}$	$\mathbb{Z}/d\mathbb{Z}$
$d = n$	$\mathbb{Z}/(dN/n)\mathbb{Z}$	$\mathbb{Z}/(dN/n)\mathbb{Z}$	$\mathbb{Z}/d\mathbb{Z}$

In both cases, $\mathrm{Br}(Q) = 0$.

4. Cohomology and Fundamental Groups

Over $k = \mathbb{C}$, the rational cohomology of the universal base $B = B_{n,d}$ is computed explicitly. The rational cohomology algebra is exterior: $$H^*(B_0; \mathbb{Q}) \cong \bigwedge_{\mathbb{Q}} \langle \xi_{2n+1}, \xi_{2n+3}, \dotsc, \xi_{2N-1} \rangle, \quad P_{B_0}(t) = \prod_{j=n+1}^{N}(1 + t^{2j-1}),$$ where $N = \dim S^d V$. For larger fibers, $B_i$, the cohomology is a tensor product with the cohomology of a Grassmannian (of codimension determined by the construction). Leray–Hirsch yields the cohomology of the universal Severi–Brauer variety as

$$H^*(Q_i; \mathbb{Q}) \cong H^*(B_i; \mathbb{Q}) \otimes H^*(\mathbb{P}^{n-1}; \mathbb{Q}).$$

These calculations position $B_{n,d}$ as a precise classifying space for Severi–Brauer varieties of the specified type.

5. Applications: Discriminant Avoidance and Period–Index Problems

One substantial application of the universal Severi–Brauer variety is to discriminant avoidance for period–index problems. The construction enables reduction of the period–index problem for ramified Brauer classes over quasi-projective varieties to the unramified (projective) case via Lefschetz-style arguments and degeneration techniques $[2510.20474]$.

Key Theorem (Discriminant Avoidance)

Suppose that for every smooth projective variety $X$ of dimension $\le m$ and every unramified Brauer class $\alpha$, the index divides the period to a fixed exponent. Then the same divisibility must hold for arbitrary Brauer classes (ramified or not) on smooth quasi-projective varieties of the same dimension, via specialization and the universality of $B_{n,d}$.

This principle is pivotal for obtaining general bounds in the period–index conjecture and for controlling the behavior of Severi–Brauer varieties in families.

6. Significance, Broader Impact, and Further Directions

The construction and study of universal Severi–Brauer varieties augment the classical understanding by providing geometric moduli spaces with prescribed period and index, facilitating global techniques—such as complete intersection and Lefschetz theorems—in addressing arithmetic and cohomological questions about division algebras. The explicit determination of their Picard, Brauer, and cohomology groups, as well as their (simple-)connectedness in most cases, positions them as effective classifying spaces for Brauer–Severi geometry. These structures unify various moduli-theoretic approaches and indicate new directions in the study of the period–index problem, the geometry of twisted forms, and the arithmetic of projective homogeneous varieties $[2510.20474]$.