Minimal Real Generalized Trisecants

Abstract

Ranestad, Seigal, and Wang asked whether, for every admissible triple (d,n,m)(d,n,m), there exists a smooth, irreducible, non-degenerate real projective variety of dimension mm, codimension nn, and degree dd, whose maximum number of real points in a transverse complementary linear section is the minimal value n+1n+1. This note observes that the question has a negative answer as stated. Proposition 2.8(2) of Jinhyung Park’s small-degree theorem excludes the underlying complex varieties for the infinite family (d,n,m)=(n+3,n,m),n≥2,m≥n+2.(d,n,m)=(n+3,n,m),\qquad n\ge2,\quad m\ge n+2. Indeed, Park’s result requires d≥2n+2d\ge2n+2 in this range, contradicting d=n+3d=n+3. The smallest resulting counterexample to universal existence is (d,n,m)=(5,2,4)(d,n,m)=(5,2,4). The obstruction is complex-geometric and arises before the condition on real complementary linear sections is considered. A revised formulation of the open problem, restricted to numerical triples for which an appropriate smooth complex projective variety exists, is proposed.