The geometry of 1-based minimal types

Abstract

In this paper, we study the geometry of a (nontrivial) 1-based <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper U"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">SU</mml:annotation> </mml:semantics> </mml:math> </inline-formula> rank-1 complete type. We show that if the (localized, resp.) geometry of the type is modular, then the (localized, resp.) geometry is projective over a division ring. However, unlike the stable case, we construct a locally modular type that is not affine. For the general 1-based case, we prove that even if the geometry of the type itself is not projective over a division ring, it is when we consider a 2-fold or 3-fold of the geometry altogether. In particular, it follows that in any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi> ω </mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -categorical, nontrivial, 1-based theory, a vector space over a finite field is interpretable.