# A bound on the geometric genus of projective varieties

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A bound on the geometric genus of projective varieties ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze JOEHARRIS Abound on the geometric genus of projective varieties Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 8, no 1 (1981), p. 35-68. <http://www.numdam.org/item?id=ASNSP_1981_4_8_1_35_0> © Scuola Normale Superiore, Pisa, 1981, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion 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/ A Bound on the Geometric Genus of Projective Varieties. JOE HARRIS 0. - Introduction. The genus of a plane curve C is readily calculated in terms of its degree and singularities by any one of a number of elementary means. The genus of a curve in Pn is not so easily described: smooth curves of a given degree in Pn may have many different genera. One question we may reasonably hope to answer, however, is to determine the greatest possible genus of an irreducible, non-degenerate curve of degree d in Pn; this problem was solved in 1889 by Castelnuovo (3), (4) (5), who went on to give a complete geometric description of those curves which achieved his bound. In this paper, we will answer the analogous question for varieties of arbitrary dimension: what is the greatest possible geometric genus of an irreducible, nondegenerate variety of dimension k and degree d in pn, We begin in section 1 by recounting Castelnuovo’s argument; a more detailed version of the argument may be found in (3) and in (4). In section 2 we use the stan

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