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Spaces of algebraic maps from real projective spaces to toric varieties. (English) Zbl 1353.55009
Given manifolds $$X$$ and $$Y$$, write $$\text{Map}^\ast(X,Y)$$ for the space of bas-point preserving continuous maps from $$X$$ to $$Y$$. If $$X$$ and $$Y$$ are complex manifolds, J. Mostovoy [Topology 45, No. 2, 281–293 (2006; Zbl 1086.58005); Q. J. Math. 63, No. 1, 181–187 (2012; Zbl 1237.58012)] determined an integer $$n_D$$ such that the inclusion map $$j_D : \text{Hol}^\ast_D(X,Y)\to \text{Map}^\ast(X,Y)$$ is a homology equivalence through dimension $$n_D$$, where $$D=(d_1,\ldots,d_r)$$ is a tuple of integers and $$\text{Hol}^\ast_D(X,Y)$$ denotes the space of holomorphic maps from $$X$$ to $$Y$$ of degree $$D$$. Recently, J. Mostovoy and E. Munguia-Villanueva [Spaces of morphisms from a projective space to a toric variety, preprint, CP^m\) to a compact smooth toric variety $$X_\Sigma$$ associated to a fan $$\Sigma$$.
Given algebraic varieties $$X$$ and $$Y$$, write $$\text{Alg}^\ast_D(X,Y)$$ for the space of algebraic (regular) maps from $$X$$ to $$Y$$ of degree $$D$$ and $$A_D(X,Y)$$ for the space of tuples of polynomials representing elements of $$\text{Alg}^\ast_D(X,Y)$$. For a real projective space $$\mathbb{R}P^m$$, the authors study the natural map $$i_D : A_D(m,X_\Sigma)\to \text{Map}^\ast(X,Y)$$ and show that the induced map $$i'_D : A_D(m,X_\Sigma;g)\to F(\mathbb{R}R^m,X_\Sigma;g)\simeq\Omega^m X_\Sigma$$ is a homology equivalence through a dimension $$n_D(d_1,\ldots,d_r;m)$$.

##### MSC:
 55R80 Discriminantal varieties and configuration spaces in algebraic topology 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 55P10 Homotopy equivalences in algebraic topology 55P35 Loop spaces
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