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**1 - 2**of**2**### ON RAMIFIED COVERS OF THE PROJECTIVE PLANE I: SEGRE’S THEORY AND CLASSIFICATION IN SMALL DEGREES WITH AN APPENDIX BY EUGENII SHUSTIN

, 903

"... Abstract. We study ramified covers of the projective plane P 2. Given a smooth surface S in P n and a generic enough projection P n → P 2, we get a cover π: S → P 2, which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as sing ..."

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Abstract. We study ramified covers of the projective plane P 2. Given a smooth surface S in P n and a generic enough projection P n → P 2, we get a cover π: S → P 2, which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. Several questions arise: First, What is the geography of branch curves among all cuspidal-nodal curves? And second, what is the geometry of branch curves; i.e., how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e., form a special 0-cycle on the plane. We start with reviewing what is known about the answers to these questions, both simple and some non-trivial results. Secondly, the classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in P 3. We give an interpretation of the work of Segre in terms of relation between Picard and Chow groups of 0-cycles on a singular plane curve B. We also review examples of small degree. In addition, the Appendix written by E. Shustin shows the existence of new Zariski pairs.

### ON COMPLETE DEGENERATIONS OF SURFACES WITH ORDINARY SINGULARITIES IN P 3

, 902

"... Abstract. We investigate the problem of existence of degenerations of surfaces in P 3 with ordinary singularities into plane arrangements in general position. Introduction. In the article we investigate degenerations of surfaces in P 3 with ordinary singularities. To begin, consider the classical pr ..."

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Abstract. We investigate the problem of existence of degenerations of surfaces in P 3 with ordinary singularities into plane arrangements in general position. Introduction. In the article we investigate degenerations of surfaces in P 3 with ordinary singularities. To begin, consider the classical prototype of this situation, namely, degenerations of plane algebraic curves. As is known, any smooth projective curve can be projected to P 2 onto a nodal curve C — a curve with ordinary