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A bound on the geometric genus of projective varieties verifying certain flag conditions


Author: Vincenzo Di Gennaro
Journal: Trans. Amer. Math. Soc. 349 (1997), 1121-1151
MSC (1991): Primary 14J99, 14M07, 14M10; Secondary 14F17
DOI: https://doi.org/10.1090/S0002-9947-97-01785-6
MathSciNet review: 1390976
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Abstract: Fix integers $n,r,s_{1},...,s_{l}$ and let $\mathcal {S}(n,r;s_{1},...,s_{l})$ be the set of all integral, projective and nondegenerate varieties $V$ of degree $s_{1}$ and dimension $n$ in the projective space $\mathbf {P}^{r}$, such that, for all $i=2,...,l$, $V$ does not lie on any variety of dimension $n+i-1$ and degree $<s_{i}$. We say that a variety $V$ satisfies a flag condition of type $(n,r;s_{1},...,s_{l})$ if $V$ belongs to $\mathcal {S}(n,r;s_{1},...,s_{l})$. In this paper, under the hypotheses $s_{1}>>...>>s_{l}$, we determine an upper bound $G^{h}(n,r;s_{1},...,s_{l})$, depending only on $n,r,s_{1},...,s_{l}$, for the number $G(n,r;s_{1},...,s_{l}):= {max} {\{} p_{g}(V) : V\in \mathcal {S}(n,r;s_{1},...,s_{l}){\}}$, where $p_{g}(V)$ denotes the geometric genus of $V$. In case $n=1$ and $l=2$, the study of an upper bound for the geometric genus has a quite long history and, for $n\geq 1$, $l=2$ and $s_{2}=r-n$, it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical data $n,r,s_{1},...,s_{l}$. For instance, we extend Halphen's theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension $n$ in $\mathbf {P}^{n+3}$ as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces in $\mathbf {P}^{5}$. Next we discuss how far $G^{h}(n,r;s_{1},...,s_{l})$ is from $G(n,r;s_{1},...,s_{l})$ and show a sort of lifting theorem which states that, at least in certain cases, the varieties $V\in \mathcal {S}(n,r;s_{1},...,s_{l})$ of maximal geometric genus $G(n,r;s_{1},...,s_{l})$ must in fact lie on a flag such as $V=V_{s_{1}}^{n}\subset V_{s_{2}}^{n+1}\subset ...\subset V_{s_{l}}^{n+l-1}\subset {\mathbf {P}^{r}}$, where $V^{j}_{s}$ denotes a subvariety of $\mathbf {P}^{r} $ of degree $s$ and dimension $j$. We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo's regularity of varieties verifying flag conditions.


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  • [ACGH] E.Arbarello, M.Cornalba, P.A.Griffiths and J.Harris, Geometry of algebraic curves, Volume I, Springer-Verlag, Berlin, 1985. MR 86h:14019
  • [BF] R.Braun and G.Fløystad, A bound for the degree of smooth surfaces in ${\mathbf {P}^{4}}$ not of general type, Compositio Mathematica 93 (1994), 211-229. MR 95g:14037
  • [Ca] G.Castelnuovo, Ricerche di geometria sulle curve algebriche, Zanichelli, Bologna, 1937.
  • [CC] L.Chiantini and C.Ciliberto, A few remarks on the lifting problem, Journées de Géométrie Algébrique d'Orsay (1992), Astérisque, no. 218, Soc. Math. France, Paris, 1993, pp. 95-109. MR 95c:14072
  • [CCD1] L.Chiantini, C.Ciliberto and V.Di Gennaro, The genus of projective curves, Duke Math. J. 70 (1993), 229-245. MR 94b:14027
  • [CCD2] L.Chiantini, C.Ciliberto and V.Di Gennaro, On the genus of projective curves verifying certain flag conditions, preprint, 1995.
  • [CCD3] L.Chiantini, C.Ciliberto and V.Di Gennaro, The genus of curves in ${\mathbf {P}^{4}}$ verifying certain flag conditions, Manuscripta Math. 88 (1995), 119-134. MR 96h:14044
  • [Ci1] C.Ciliberto, Hilbert functions on finite sets of points and the genus of a curve in a projective space, Space Curves: Proceedings, Rocca di Papa, 1985, Lecture Notes in Math., vol. 1266, Springer-Verlag, Berlin, 1987, pp. 24-73. MR 89c:14039
  • [Ci2] C.Ciliberto, On a property of Castelnuovo varieties, Transactions of the American Mathematical Society 303/1 (1987), 201-210. MR 88e:14055
  • [Ci3] C.Ciliberto, Alcune applicazioni di un classico procedimento di Castelnuovo, Seminari di Geometria 1982-1983, Università di Bologna, Bologna, 1984, pp. 17-43. MR 86j:14021
  • [D] V.Di Gennaro, Generalized Castelnuovo varieties, manuscripta math. 81 (1993), 311-328. MR 95f:14074
  • [EH] D.Eisenbud and J.Harris, Curves in projective space, Sém. Math. Sup., vol. 85, Les Presses de l'Université de Montréal, Montréal, vol. 85, 1982. MR 84g:14024
  • [E] P.Ellia, Sur les lacunes d'Halphen, Algebraic Curves and Projective Geometry: Proceedings, Trento, 1988, Lecture Notes in Math., vol. 1389, Springer-Verlag, Berlin, 1989, pp. 43-65. MR 90j:14031.
  • [EP] G.Ellingsrud and C.Peskine, Sur les surfaces lisses de ${\mathbf {P}_{4}}$, Invent. Math. 95 (1989), 1-11. MR 89j:14023
  • [GH] P.A.Griffiths and J.Harris, Principles of algebraic geometry, Wiley, New York, 1978. MR 80b:14001
  • [GLP] L.Gruson, R.Lazarsfeld and C.Peskine, On a Theorem of Castelnuovo, and the Equations Defining Space Curves, Invent. Math. 72 (1983), 491-506.
  • [GP] L.Gruson and C.Peskine, Genre des courbes dans l'espace projectif, Algebraic Geometry: Proceedings, Norway, 1977, Lecture Notes in Math., vol. 687, Springer Verlag, New York, 1978, pp. 31-59. MR 81e:14019
  • [Ha] G.Halphen, Mémoire sur la classification des courbes gauches algébriques, Oeuvres complètes t.III.
  • [Hr1] J.Harris, The genus of space curves, Math. Ann. 249 (1980), 191-204. MR 81i:14022
  • [Hr2] J.Harris, A bound on the geometric genus of projective varieties, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981), 35-68. MR 82h:14010
  • [Ht] R.Hartshorne, Algebraic geometry, Springer-Verlag, Berlin, 1977. MR 57:3116
  • [J] F.Jongmans, Contribution à la théorie des variétés algébriques, Thèse d'Agrégation de l'Einseignement supérieur, Bruxelles, M.Hayez, Imprimeur de l'Académie Royale de Belgique 1947; also published as Mém. Soc. Roy. Sci. Liège (4) 7 (1947), 367-468. MR 9:611c
  • [K] E.Kunz, Holomorphe Differenzialformen auf algebraischen Varietäten mit Singularitäten. I, Manuscripta Math. 15 (1975), 91-108. MR 52:5674
  • [M] D.Mumford, Lectures on Curves on an Algebraic Surface, Ann. of Math. Stud., Princeton Univ. Press, Princeton, N.J., 1966. MR 35:187
  • [N] U.Nagel, On bounds for cohomological Hilbert functions, notes from author's thesis ``Über Gradschranken für Syzygien und kohomologische Hilbertfunktionnen'', Paderborn, 1990, J. Algebra 150 (1992), 231-244. MR 93i:13011
  • [NV] U.Nagel and W.Vogel, Bounds for Castelnuovo's regularity and Hilbert functions, Topics in Algebra, Banach Center Publ., vol. 26; part 2, PWN, Warsaw (1990), pp. 163-183. MR 93k:13024
  • [No] M.Noether, Zur Grundlegung der Theorie der algebraischen Raumcurven, Königlichen Akad. der Wissenschaften, 1883.
  • [PS] C.Peskine and L.Szpiro, Liaison des variétés algébriques. I, Inventiones Math. 26 (1974), 271-302. MR 51:526
  • [Re] R.Re, Sulle sezioni iperpiane di una varietà proiettiva, Le Matematiche (Catania) 42 (1987), 211-218 (1989). MR 90j:14069
  • [R] M.Reid, Canonical 3-folds, Journées de Géométrie Algébrique d'Angers 1979, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, pp. 273-310. MR 82i:14025.
  • [Se] E.Sernesi, Topics on families of projective schemes, Queen's Papers Pure Appl. Math., vol. 73, Queen's Univ., Kingston, Ont., 1986. MR 88b:14006
  • [Sv] F.Severi, Intorno ai punti doppi improprii di una superficie generale dello spazio a 4 dimensioni e a suoi punti tripli apparenti, Rendiconti di Palermo 15 (1901).
  • [St] R.Strano, On generalized Laudal's lemma, Complex Projective Geometry, (Trieste and Bergen, 1989; G. E. Ellingsrud et al., eds.), London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Unov. Press, Cambridge, 1992, pp. 284-293. MR 94a:14032

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Additional Information

Vincenzo Di Gennaro
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata", 00133 Roma, Italy
Email: Digennaro@mat.utovrm.it

DOI: https://doi.org/10.1090/S0002-9947-97-01785-6
Keywords: Projective varieties, geometric genus, arithmetic genus, flag conditions, Castelnuovo's theory, low codimension varieties, complete intersections, Castelnuovo's regularity
Received by editor(s): October 23, 1995
Article copyright: © Copyright 1997 American Mathematical Society

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