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On the space of holomorphic maps from the Riemann sphere to the quadric cone. (English) Zbl 0802.58012
Let $$M_ 2$$ be the quadric cone $$\{[z_ 0; z_ 1; z_ 2; z_ 3] \in \mathbb{C} P^ 3/z^ 2_ 2 = z_ 1z_ 3\}$$. The theme of this paper is the comparison of the homology groups of $$\text{Hol}_ d (S^ 2, M_ 2)$$, the space of holomorphic maps $$S^ 2 \to M_ 2$$ of degree $$d$$, with that of $$\text{Map}_ d (S^ 2, M_ 2)$$, the space of continuous functions of degree $$d$$. For example it is shown that the inclusion $$\text{Hol}^*_ d (S^ 2,M_ 2) \to \text{Map}_ d^* (S^ 2,M_ 2)$$ induces an isomorphism on homology groups of dimension $$<[d/3]$$ and a surjection at dimension $$[d/3]$$, where the $$*$$ denotes maps $$f$$ for which $$f(\infty) = [1;1;1;1]$$.

##### MSC:
 58D15 Manifolds of mappings 32C18 Topology of analytic spaces
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