Geodesic
Remarks on the Nagata Conjecture
Serdica Mathematical Journal, Tome 30 (2004) no. 2-3, pp. 405-430.

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The famous Nagata Conjecture predicts the lowest degree of a plane curve passing with prescribed multiplicities through given points in general position. We explain how this conjecture extends naturally via multiple point Seshadri constants to ample line bundles on arbitrary surfaces. We show that if there exist curves of unpredictable low degree, then they must have equal multiplicities in all but possibly one of the given points. We use this restriction in order to obtain lower bounds on multiple point Seshadri constants on a surface. We discuss also briefly a seemingly new point of view on the Nagata Conjecture via the bigness of the involved linear series.
Keywords: Nagata Conjecture, Linear Series, Seshadri Constants, Harbourne-Hirschowitz Conjecture, Big Divisors 
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Strycharz-Szemberg, Beata; Szemberg, Tomasz. Remarks on the Nagata Conjecture. Serdica Mathematical Journal, Tome 30 (2004) no. 2-3, pp. 405-430. http://geodesic.mathdoc.fr/item/SMJ2_2004_30_2-3_a15/