Bulletin of the
Korean Mathematical Society BKMS

Let $K$ be a fan-like simplicial sphere of dimension $n-1$ such that its associated complete fan is strongly polytopal, and let $v$ be a vertex of $K$. Let $K(v)$ be the simplicial wedge complex obtained by applying the simplicial wedge operation to $K$ at $v$, and let $v_0$ and $v_1$ denote two newly created vertices of $K(v)$. In this paper, we show that there are infinitely many strongly polytopal fans $\Sigma$ over such $K(v)$'s, different from the canonical extensions, whose projected fans ${\rm Proj}_{v_i} \Sigma$ $(i=0,1)$ are also strongly polytopal. As a consequence, it can be also shown that there are infinitely many projective toric varieties over such $K(v)$'s such that toric varieties over the underlying projected complexes $K_{{\rm Proj}_{v_i} \Sigma}$ $(i=0,1)$ are also projective.

Keywords: simplicial complexes, strongly polytopal, simplicial wedge operation, projective toric varieties, linear transforms, Gale transforms, Shephard's diagrams, Shephard's criterion

MSC numbers: 14M25, 52B20, 52B35