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Chain-connected component decomposition of curves on surfaces. (English) Zbl 1193.14047

From the Introduction: “In the study of algebraic surfaces, we often encounter reducible non-reduced curves. Typical examples are various cycles supported by the exceptional set of a normal surface singularity and similar fibres in a fibred surface. Needless to say, any reducible curve decomposes into a sum of irreducible curves uniquely up to the order. As one may see from the success of 1-connected curves C. P. Ramanujam [J. Indian Math. Soc. 36, 41–51, (1972; Zbl 0276.32018) and 38, 121–124 (1974; Zbl 0368.14005)], E. Bombieri [Publ. Math. I.H.É.S. 42, 171–219 (1973; Zbl 0339.14026)], however, it is sometimes more convenient and even natural to treat a connected reducible curve as if it were a single irreducible curve. In other words, some coarser decompositions could be better suited to certain problems than the decomposition into irreducible components.
The purpose of the paper is to revive and recast another canonical way to decompose reducible curves on a smooth surface used by Y. Miyaoka [Journées de géométrie algèbrique d’Angers, 1979 (ed. A. Beauville), Sijthoff & Noordhoff, 239–247, (1980; Zbl 0446.14007)]. Our atomic curves are chain-connected curves (what Miyaoka called s-connected divisors), which are themselves generally reducible. The decomposition theorem (Corollary 1.7) states that every effective divisor on a smooth surface decomposes into a sum of chain-connected curves enjoying nice numerical relations. Such an orderly decomposition is also essentially unique. We call it a chain-connected component decomposition. We know that 1-connectivity is a very important notion in surface theory, but the class of 1-connected curves is not large enough to cover certain important classes, such as fundamental cycles of singularities. Chain-connectivity is a notion that dates back to K. Kodaira [J. Math. Soc. Japan 20, 170–192 (1968; Zbl 0157.27704)], it is defined by a weaker condition and it covers a considerably wider range.”
The last section of the paper under review contains results concerning fibred surfaces, which the author studied further in another paper [Tohoku Math. J. (2) 62, No. 1, 117–136 (2010; Zbl 1193.14048)].

MSC:

14J25 Special surfaces
14H45 Special algebraic curves and curves of low genus
14J29 Surfaces of general type
14J17 Singularities of surfaces or higher-dimensional varieties
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