Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝔤</m:mi> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>𝔤</m:mi> <m:mover accent="true"> <m:mn>0</m:mn> <m:mo stretchy="false">¯</m:mo> </m:mover> </m:msub> <m:mo>⊕</m:mo> <m:msub> <m:mi>𝔤</m:mi> <m:mover accent="true"> <m:mn>1</m:mn> <m:mo stretchy="false">¯</m:mo> </m:mover> </m:msub> </m:mrow> </m:mrow> </m:math> {{g}={g}0̄⊕{g}1̄} be a basic classical Lie superalgebra over an algebraically closed field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝐤</m:mi> </m:math> {{k}} of characteristic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {p>2} . Denote by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒵</m:mi> </m:math> {Z} the center of the universal enveloping algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>U</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>𝔤</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {U({g})} . Then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒵</m:mi> </m:math> {Z} turns out to be finitely-generated purely-even commutative algebra without nonzero divisors. In this paper, we demonstrate that the fraction <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Frac</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">𝒵</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Frac(Z)} is isomorphic to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Frac</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ℨ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Frac(Z)} for the center <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℨ</m:mi> </m:math> {Z} of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>U</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>𝔤</m:mi> <m:mover accent="true"> <m:mn>0</m:mn> <m:mo stretchy="false">¯</m:mo> </m:mover> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {U({g}0̄)} . Consequently, both Zassenhaus varieties for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔤</m:mi> </m:math> {{g}} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝔤</m:mi> <m:mover accent="true"> <m:mn>0</m:mn> <m:mo stretchy="false">¯</m:mo> </m:mover> </m:msub> </m:math> {{g}0̄} are birationally equivalent via a subalgebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mover accent="true"> <m:mi mathvariant="script">𝒵</m:mi> <m:mo>~</m:mo> </m:mover> <m:mo>⊂</m:mo> <m:mi mathvariant="script">𝒵</m:mi> </m:mrow> </m:math> {{Z}} , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Spec</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">𝒵</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Spec(Z)} is rational under the standard hypotheses.