2018/06/28 by Daigle, Daniel
#14M20 #14R05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary: 14R10. Secondary: 14R20
paper · doi:10.48550/arxiv.1806.10739
Let k be a field of characteristic zero and B a commutative integral domain that is also a finitely generated k-algebra. It is well known that if k is algebraically closed and the "Field Makar-Limanov" invariant FML(B) is equal to k, then B is unirational over k. This article shows that, when k is not assumed to be algebraically closed, the condition FML(B)=k implies that there exists a nonempty Zariski-open subset U of Spec(B) with the following property: for each prime ideal \mathfrakp ∈ U, the κ(\mathfrakp)-algebra κ(\mathfrakp) ⊗k B can be embedded in a polynomial ring in n variables over κ(\mathfrakp), where n=dim B and κ(\mathfrakp) = B_\mathfrakp/\mathfrakpB_\mathfrakp.