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CORRECTIONS to be made to the article by S. Kleiman and R. Piene ENUMERATING SINGULAR CURVES ON SURFACES appearing in "Algebraic geometry -Hirzebruch 70,"Cont. Math. 241 (1999), 209-238 p. 217b. -In the displayed formula for cod(D), replace 'm D ' by 'm V '. p. 219. -In Table 2-1, the value of r for X 1,2 should not be 1, but 4.p. 221b. -The proof of Proposition (3.2) should have used Gotzmann's regularity theorem in much the same way that it is used in the proof of Proposition (3.5). So replace the second paragraph in the first proof by the following two. There exists a map ϕ from the germ of Y at y into B carrying y to the origin. Since m ≥ µ − 1 by hypothesis, ϕ is smooth, as we'll now show. It suffices to show the surjectivity of the map of tangent spaces, which is equal to the natural map,Since N is spanned, it suffices to show the surjectivity of the map,To show the surjectivity of (3.2.1), embed S in a projective space P so that M = O S (1), and let K C,P ⊂ O P be the ideal such thatHence, by Gotzmann's regularity theorem [13] (see also [14, p. 80]), the ideal K C,P is µ-regular. So H 1 (K C,P (m)) vanishes for m ≥ µ − 1. Hence the mapIn the third paragraph of the proof of (3.7), replace the third sentence by the following one. On the other hand, every fiber of Z(D) → H(D) is a projective space, and meets Z 0 (D) in a nonempty open subset by (3.5). p. 226m. -In the second paragraph of Section 4, replace the first clause of the first sentence by the following one. Let π: F → Y be a smooth and projective family of (possibly reducible) surfaces, where Y is equidimensional and Cohen-Macaulay, and . . . 1 p. 226m. -In the displayed sequence of principal parts, the the first term should be twisted by D too: 0 → Sym i−1 Ω 1 F/Y (D) → · · · . p. 230t. -In (4.5), replace the final '=' by '≤', getting . . . a + b + 2c ≤ r + 2. (4.5)p. 230m. -In the paragraph that begins, "The genericity hypothesis also implies," replace the first sentence by the following one.The genericity hypothesis also implies that X 2 is reduced, Cohen-Macaulay, and equidimensional of codimension 3 in F . p. 231. -In the second display, replace w 1 − e by w 1 + e, and w 2 + e 2 by w 2 − e 2 . In the third display, replace w 1 − e by w 1 + e, and w 2 + e by w 2 − e 2 . In the next to the last paragraph, replace w 1 − e by w 1 + e and w 2 + e by w 2 − e 2 , and w 3 2 by w 2 e. In the display, remove [X i ], and move the sentence following the display down to after the next display. p. 237. -References 3, 4, 5, and 8 have appeared. The bibliographic data follows. 3.Abstract. We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and up to three nodes. The curves must also pass through appropriately many general points. The number of curves is given by a universal polynomial in four basic Chern numbers.To justify the enumeration, we make a rudimentary classification of ...

Polar classes of singular varieties Annales scientifiques de l'É.N.S. 4 e série, tome 11, n o 2 (1978), p. 247-276 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1978, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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We continue the development of methods for enumerating nodal curves on smooth complex surfaces, extending the range of validity. We apply the new methods in three important cases. First, for up to eight nodes, we prove Göttsche's conjecture about plane curves of low degree. Second, we prove Vainsencher's conjectural enumeration of irreducible six-nodal plane curves on a general quintic threefold in four-space, which is important for Clemens' conjecture and mirror symmetry. Third, we supplement Bryan and Leung's enumeration of nodal curves in a given homology class on an Abelian surface of Picard number 1.