Monodromy of rational curves on \(K3\) surfaces of low genus. (English) Zbl 1485.14020
Summary: In many situations, the monodromy group of enumerative problems will be the full symmetric group. In this paper, we study a similar phenomenon on the rational curves in \(| \mathcal{O}(1) |\) on a generic \(K3\) surface of fixed genus over \(\mathbb{C}\) as the \(K3\) surface varies. We prove that when the \(K3\) surface has genus \(g\), \(1 \leq g \leq 3\), the monodromy group is the full symmetric group.
MSC:
| 14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |
| 14J20 | Arithmetic ground fields for surfaces or higher-dimensional varieties |
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
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