Timeline for Branch loci of Ramified covers

5 events

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Nov 27, 2013 at 14:32	comment	added	Mohammad Farajzadeh-Tehrani		That's fine, what I care about is the underlying reduced divisor. I want the reduced branch locus to be normal crossing, otherwise you are absolutely right. May be I should have said that in my question.
Nov 27, 2013 at 14:22	comment	added	Francesco Polizzi		In other words, your example is a deformation of the general cover, in which the six-cuspidal sextic degenerates to a smooth cubic counted twice.
Nov 27, 2013 at 14:19	comment	added	Francesco Polizzi		This is very classical, Zarisky surely knew this. For a modern treatment and other examples, see Miranda's paper Triple covers in Algebraic Geometry, section 10. In your example the schematic branch locus must be counted with multiplicity two, since there is total ramification. So it is not reduced, hence not normal crossing. It must be a sextic curve in any case.
Nov 27, 2013 at 14:14	comment	added	Mohammad Farajzadeh-Tehrani		Very nice example. Could please recall one reference for this example. By the way, Fermat cubic surface in $\mathbb{P}^3$, $x_0^3+x_1^3+x_2^3+x_3^3=0$ via the map $[x_0,x_1,x_2,x_3]\to [x_1,x_2,x_3]$ is cyclic cover of $\mathbb{P}^2$ ramified along a smooth cubic curve. It is not obvious that the covering you are mentioning is not deformable to this one.
Nov 27, 2013 at 12:15	history	answered	Francesco Polizzi	CC BY-SA 3.0