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Monodromy of projective curves. (English) Zbl 1084.14011

It is known that for a general curve \(X\) of genus \(g>3\) the monodromy group of a degree \(d\) indecomposable covering \(X\rightarrow {\mathbb P}^1\) is either \(S_d\) (in this case the covering is said to be uniform) or \(A_d\). The uniform position principle states that, given an irreducible non-degenerate curve \(C\subset {\mathbb P}^r({\mathbb C})\), a general \((r-2)\)-plane \(L\subset {\mathbb P}^r\) is uniform, i.e. projection from \(L\) induces a rational map whose monodromy group is \(S_d\). The authors show that the locus of non-uniform \((r-2)\)-planes has codimension at least two in the Grassmannian. The result is sharp because if there is a point \(x\in {\mathbb P}^r\) such that the projection from \(x\) induces a map which is not birational onto its image, then the Schubert cycle \(\sigma (x)\) of \((r-2)\)-planes through \(x\) is contained in the locus of non-uniform \((r-2)\)-planes. For a smooth curve \(C\subset {\mathbb P}^3\) they show that any irreducible surface of non-uniform lines is a cycle \(\sigma (x)\) as above, unless \(C\) is a rational curve of degree three, four or six.

MSC:

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14H99 Curves in algebraic geometry
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