--- title: Cremona Dimension and Birational Group Actions url: https://www.emergentmind.com/topics/cremona-dimension type: topic --- # Cremona Dimension and Birational Group Actions Searching arXiv for recent papers on “Cremona dimension” and closely related formulations. “Cremona dimension” most commonly denotes the minimal dimension of a rational variety on which a group acts faithfully by birational transformations, equivalently the minimal \(n\) such that the group embeds into the \(n\)-dimensional Cremona group \(Cr_k(n)=\operatorname{Bir}(\mathbb P^n_k)\) [2507.15096]. In the recent literature, however, the phrase also appears in adjacent but non-equivalent senses: as a dimension-sensitive extension problem for finite subgroups of low-rank Cremona groups, as a representation-theoretic uniform bound for finite subgroups of \(\mathrm{Cr}_n(k)\), and as a broader label for questions about the global structure of Cremona groups that are not captured by ordinary algebraic-group dimension [2510.13200]. The subject is therefore best understood as a family of dimension-based invariants and classification problems centered on birational actions of groups on rational or rationally connected varieties. ## 1. Formal definition and basic inequalities Over an algebraically closed field \(k\) of characteristic \(p\ge 0\), the Cremona dimension of a finite group \(G\) is defined by \[ crd_k(G)=\min\{n:\ G\ \text{embeds in } Cr_k(n)\}, \] where \[ Cr_k(n)=\operatorname{Bir}(\mathbb P^n_k)=\operatorname{Aut}(k(t_1,\dots,t_n)/k). \] Since every \(n\)-dimensional rational variety is birational to \(\mathbb P^n\), this is equivalent to asking for the minimal dimension of a rational variety admitting a faithful birational \(G\)-action [2507.15096]. The same survey introduces the rationally connected dimension \[ rcd_k(G)=\min\{\dim X:\ X \text{ is a rationally connected }G\text{-variety over }k\}, \] and recalls the inequalities \[ rcd_k(G)\le ed_k(G),\qquad rcd_k(G)\le crd_k(G), \] where \(ed_k(G)\) is the essential dimension. It also records the basic formal properties \[ crd_k(H)\le crd_k(G)\quad\text{if }H\subset G, \] and \[ crd_k(G_1\times G_2)\le crd_k(G_1)+crd_k(G_2). \] For normal subgroups \(N\triangleleft G\), one has \[ crd_k(G)\ge rcd_k(G/N). \] These statements place Cremona dimension between linear representation theory and birational geometry: it is birational rather than linear, but still monotone under subgroup formation and subadditive under direct products [2507.15096]. A central conjectural comparison is \[ crd_\mathbb C(G)\le ed_\mathbb C(G), \] proposed for finite groups over \(\mathbb C\). The available examples assembled in the survey support this inequality, often strictly [2507.15096]. ## 2. Low-dimensional values and explicit group-theoretic examples The low-dimensional range supplies the main evidence for the basic theory. The survey records that \(ed_\mathbb C(G)=0\) only for the trivial group, and if \(ed_\mathbb C(G)=1\), then \(G\) is cyclic or dihedral of order \(2(2k+1)\); such groups lie in \(Cr_\mathbb C(1)=PGL_2(\mathbb C)\), hence satisfy \(crd_\mathbb C(G)\le 1=ed_\mathbb C(G)\). For \(ed_\mathbb C(G)=2\), Duncan’s classification implies that all such groups occur in \(Cr_\mathbb C(2)\), so \(crd_\mathbb C(G)\le 2\) in that range [2507.15096]. Abelian and \(p\)-group examples show that Cremona dimension can be much smaller than essential dimension. For finite \(p\)-groups in characteristic different from \(p\), the Karpenko–Merkurjev theorem implies that \(ed_k(P)\) equals the minimal dimension of a faithful linear representation, and therefore \[ crd_k(P)\le ed_k(P). \] For elementary abelian \(2\)-groups, the survey uses the fact that \[ 2^{2s}\cong \mathfrak D_4^s \] acts faithfully on \((\mathbb P^1)^s\), obtaining \[ crd_k(2^r)\le \Bigl[\frac r2\Bigr]. \] Combined with the Kollár–Zhuang lower bound \[ rcd_k(G)\ge \frac{p-1}{p}\operatorname{rank}(G), \] this yields \[ crd_k(2^r)=rcd_k(2^r)1\), the inequality is strict. The paper also records sharper special cases, including \[ crd_\mathbb C(3_+^{1+3})=2. \] Pseudo-reflection groups give further evidence. For example, the survey records \[ crd_k(W(E_6))=ed_k(W(E_6))=4, \] while for several Coxeter and Shephard–Todd groups one has strict inequality, such as \[ crd_k(F_4)=3,\ ed_k(F_4)=4,\qquad crd_k(H_4)=30\) yields such an exact sequence, splitting the group into base and fiber parts. The main question is whether every such extension is actually a direct product, because that determines whether the group can be realized on a terminal Fano variety of the same dimension [2510.13200]. The results are dimension-specific. For \(n=1\), \[ \mathcal A_1=\left\{ \mathbb Z/k \text{ for }k\ge 1,\; (\mathbb Z/2)^2 \right\}. \] For \(n=2\), Blanc’s classification gives \[ \mathcal A_2= \left\{ \mathbb{Z}/k\times \mathbb{Z}/m,\ \mathbb{Z}/2k\times (\mathbb{Z}/2)^2,\ (\mathbb{Z}/4)^2\times \mathbb{Z}/2,\ (\mathbb{Z}/3)^3,\ (\mathbb{Z}/2)^4 \right\}, \] and \(\mathcal A_2=\mathcal B_2\). Moreover, \[ \mathcal A_1\times \mathcal A_1=\mathcal A_1\bullet \mathcal A_1,\qquad \mathcal A_1\times \mathcal A_2=\mathcal A_1\bullet \mathcal A_2. \] Thus in dimensions \(2\) and \(3\), the relevant abelian extensions split [2510.13200]. For dimension \(4\), the main new theorem is \[ (\mathcal A_2 \bullet \mathcal A_2)\setminus (\mathcal A_2 \times \mathcal A_2)=\{ (\mathbb Z/4)^5 \}. \] In strengthened form, \[ (\mathcal{A}_1 \bullet \mathcal{A}'_3 + \mathcal{A}_2 \bullet \mathcal{A}_2) \setminus (\mathcal{A}_1 \times \mathcal{A}'_3 + \mathcal{A}_2 \times \mathcal{A}_2) = \{ (\mathbb{Z}/4)^5 \}. \] So on the Cremona side, all relevant abelian extensions up to dimension \(4\) split except the single non-split isomorphism type \((\mathbb Z/4)^5\). On the rationally connected side, however, the paper proves \[ \mathcal{B}_1 \bullet \mathcal{B}_3' + \mathcal{B}_2 \bullet \mathcal{B}_2 = \mathcal{B}_1 \times \mathcal{B}_3' + \mathcal{B}_2 \times \mathcal{B}_2. \] So for the currently expected \(3\)-fold class \(\mathcal B_3'\), even this obstruction disappears [2510.13200]. This dimension-sensitive extension theory is not a definition of Cremona dimension in the sense of \(crd_k(G)\), but it studies how finite groups are assembled from lower-dimensional birational actions and how that assembly changes at the threshold \(n=4\). ## 5. Structural and topological uses of “dimension” for Cremona groups Another strand of the literature uses “dimension” not for a group invariant \(crd_k(G)\), but for the large-scale structure of \(\mathrm{Cr}_n(k)\) itself. One paper proves that for \(n\ge 2\), the Cremona group cannot be endowed with the structure of an algebraic variety of infinite dimension satisfying the expected functorial universal property for algebraic families. More precisely, there is no ind-variety or ind-group structure of the expected type on \(\mathrm{Cr}_n(k)\), and the obstruction is topological [1210.6960]. The basic filtration is by degree: \[ \mathrm{Aut}(\mathbb P^n)=\mathrm{Bir}(\mathbb P^n)_{\le 1}\subset \mathrm{Bir}(\mathbb P^n)_{\le 2}\subset \mathrm{Bir}(\mathbb P^n)_{\le 3}\subset \cdots, \] where \(\mathrm{Bir}(\mathbb P^n)_{\le d}\) is the set of birational maps of degree at most \(d\). For each \(d\), the parameter space \(H_d\) of degree-\(d\) birational maps is an algebraic variety, and \[ \pi_d:H_d\to \mathrm{Bir}(\mathbb P^n)_{\le d} \] is a surjective, continuous, closed topological quotient map. Exact-degree strata \(\mathrm{Bir}(\mathbb P^n)_d\) are algebraic varieties, but bounded-degree pieces \(\mathrm{Bir}(\mathbb P^n)_{\le d}\) for \(d,n\ge 2\) are not algebraic varieties in the required sense because of degree-dropping degenerations [1210.6960]. The same paper introduces the Euclidean topology on \(\mathrm{Cr}_n(k)\) for local fields \(k\), making it a Hausdorff topological group. In that topology, inversion is a homeomorphism and composition is continuous, but for \(n\ge 2\) the group is not locally compact and not metrisable. Complementing this, a further paper proves topological simplicity: for each infinite field \(k\) and \(n\ge 1\), \(\Bir(\mathbb P^n_k)\) is topologically simple in the Zariski topology, and for each local field \(k\) and \(n\ge 2\), it is topologically simple in the Euclidean topology [1511.08907]. This body of work suggests a recurring distinction. The numerical invariant \(crd_k(G)\) measures the smallest birational ambient dimension required for a given finite group; by contrast, the global Cremona group \(\mathrm{Cr}_n(k)\) is not itself controlled by any naive infinite-dimensional algebraic-group structure. The “dimension” of the Cremona group in this structural sense is therefore better understood through degree filtrations, topology, and subgroup geometry than through an ind-group dimension theory [1210.6960]. ## 6. Related birational complexity notions and scope Several nearby notions quantify birational complexity in ways that sometimes overlap with, but do not coincide with, Cremona dimension. One is the categorical dimension of a birational map. Over a field of characteristic \(0\), the paper on categorical dimension defines \[ \operatorname{mcd}(\phi)= \min_{(b_1,c_1,\dots,b_r,c_r)} \max_i \operatorname{mcd}(C_i), \] using weak factorization and a filtration on the Grothendieck ring \(PT(k)\) of smooth proper pretriangulated dg categories [1701.06803]. For \(X=\mathbb P^n\), this yields a filtration \[ \mathrm{Bir}_{-1}(\mathbb P^n)\subseteq \mathrm{Bir}_0(\mathbb P^n)\subseteq \cdots \subseteq \mathrm{Bir}_{n-2}(\mathbb P^n)=\mathrm{Bir}(\mathbb P^n), \] which is a filtration of the Cremona group by birational maps of bounded categorical complexity [1701.06803]. Another nearby invariant is minimal Cremona degree for hypersurfaces. For a hypersurface \(X\subset \mathbb P^r\), the minimal Cremona degree is the minimal degree among all hypersurfaces Cremona equivalent to \(X\). For quartic surfaces in \(\mathbb P^3\), the possibilities are exactly \[ 1,\ 3,\ 4, \] with degree \(1\) precisely for rational quartics, degree \(3\) precisely for elliptic ruled quartics, and degree \(4\) otherwise [2105.12448]. Related work proves that every irreducible reduced rational quartic surface in \(\mathbb P^3\) is Cremona equivalent to a plane [1905.03976]. These are invariants of embedded divisors rather than of groups, but they belong to the same birational complexity landscape. A broader conclusion is therefore warranted. In the current literature, “Cremona dimension” has a precise established meaning for finite groups as \(crd_k(G)\), but adjacent papers use dimension in several other technically important ways: fixed-rank representation bounds \(c_n(k)\), dimension-sensitive extension theory for finite subgroups, and filtrations of Cremona groups by categorical or embedding complexity [2507.15096]. This suggests that the subject is less a single invariant than a small hierarchy of birational dimension theories attached to groups, maps, and embeddings inside the Cremona framework. Source: https://www.emergentmind.com/topics/cremona-dimension