UNIFORM BOUNDS FOR FIELDS OF DEFINITION IN PROJECTIVE SPACES

The paper proves the existence of a uniform bound C_n for fields of definition of algebraic structures on P^n with finite automorphism group, via a careful stack/gerbe analysis and a case-by-case argument that reduces to finding liftable rational points on a “compression” P and uses group-theoretic bounds; in the trivial-automorphism case it recovers the classical n+1 bound by splitting a Brauer–Severi variety . The model’s proof asserts a universal PGL_{n+1}-torsor over the field of moduli and concludes that splitting a single Severi–Brauer variety always suffices (yielding C_n = n+1). That step is invalid in general: pushing the residual gerbe to B PGL produces a PGL-torsor over the residual gerbe (a finite gerbe), not over Spec(k_ξ). Consequently, the obstruction is not just a Brauer class; the paper shows that descent to P^n_k is equivalent to the existence of a k-point of the universal projective bundle over the gerbe (equivalently, a liftable point on the compression P), and handling nontrivial automorphism groups requires additional bounded extensions beyond merely splitting a Brauer–Severi variety . The paper’s argument is coherent and complete; the model’s key reduction misidentifies the locus of the torsor and overstates the bound.