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Monodromy of projections. (English) Zbl 1027.14006

From the paper: Let \(X\subset \mathbb{P}^r\) be a projective variety of dimension \(n\) and degree \(d\) over the complex numbers. For each finite linear projection \(\pi: X\to \mathbb{P}^n\) let \(M(\pi)\) be the monodomy group of \(\pi\), a subgroup of the symmetric group \(\mathbb{S}_d\), defined up to conjugacy. Also, we denote by \(M(X)\) the finite collection of all \(M(\pi)\) for varying \(\pi\). This is a discrete invariant associated to \(X\).
The first two sections of this article contain preliminaries about the definition of \(M(X)\).
In §3 we sketch Lazarsfeld’s application to monodromy groups of linear series of the case of which \(X\) is a Grassmannian in its Plücker imbedding. This case was the main motivation of our work.
Section four is a report on an article by O. Zariski [Rend. Circ. Mat. Palermo 50, 196-218 (1926; JFM 52.0653.01)]. In that article Zariski classifies the projections of a rational normal curve such that the monodromy group is a Frobenius group. Main result:
Theorem 4.22. Let \(f:\mathbb{P}^1\to \mathbb{P}^1\) be a map of degree \(d\) such that the monodromy group is a Frobenius group. Then there exists a curve \(\widetilde{X}\) and subgroups \(M_1\subset M\subset \text{Aut} (\widetilde{X})\) such that \(f:\mathbb{P}^1= \widetilde{X}/M_1\to \mathbb{P}^1= \widetilde{X}/M\).

MSC:

14E05 Rational and birational maps

Citations:

JFM 52.0653.01
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