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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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