ABSTRACT
This is a report about our joint paper with my brother, to be published in Izvestiya RAN: Ser.Mat, 64:6 (2000) (see also math.AG/00021-74). We are interested in generic coverings f : X → ℙ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203912089/10e95e2c-eca6-48d5-adf5-58298374b4f7/content/eq2207.tif"/> of projective plane by surfaces X with A-D-E-singularities (in other terminology, rational double points, Du Val singularities and etc.). A classical result is that a generic projection of a non-singular surface X ⊂ ℙ r https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203912089/10e95e2c-eca6-48d5-adf5-58298374b4f7/content/eq2208.tif"/> to ℙ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203912089/10e95e2c-eca6-48d5-adf5-58298374b4f7/content/eq2209.tif"/> is a finite covering with at most folds and pleats as singularities, and the discriminant (= branch) curve B ⊂ ℙ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203912089/10e95e2c-eca6-48d5-adf5-58298374b4f7/content/eq2210.tif"/> is cuspidal. A fold (resp., a pleat) is a singularity of a map locally equivalent to a projection of a surface, defined by equation x = z 2(resp. y = z 3 + xz), to the x, y-plane. A cuspidal curve is a curve, which has at most ordinary nodes and cusps (locally defined by equation xy = 0 and y 2 = x 3). Over a node b ∈ B there are two folds, and over a cusp — one pleat. A generic(or simple) covering is a covering, which possesses the same properties as a generic projection. First of all, I explain why it is important to consider coverings by surfaces with A-D-E-singularities.
