Abstract
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 <TEX>$v_0$</TEX> and <TEX>$v_1$</TEX> denote two newly created vertices of K(v). In this paper, we show that there are infinitely many strongly polytopal fans <TEX>${\Sigma}$</TEX> over such K(v)'s, different from the canonical extensions, whose projected fans <TEX>${Proj_v}_i{\Sigma}$</TEX> (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 <TEX>$K_{{Proj_v}_i{\Sigma}}$</TEX> (i = 0, 1) are also projective.
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