Moduli of nodal curves on smooth surfaces of general type
Author:
F. Flamini
Journal:
J. Algebraic Geom. 11 (2002), 725-760
DOI:
https://doi.org/10.1090/S1056-3911-02-00322-3
Published electronically:
June 10, 2002
MathSciNet review:
1910265
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Abstract |
References |
Additional Information
Abstract: In this paper we focus on the problem of computing the number of moduli of the so called Severi varieties (denoted by $V_{| D |, \delta }$), which parametrize universal families of irreducible, $\delta$-nodal curves in a complete linear system $|D|$, on a smooth projective surface $S$ of general type. We determine geometrical and numerical conditions on $D$ and numerical conditions on $\delta$ ensuring that such a number coincides with $dim(V_{| D |, \delta })$. As related facts, we also determine some sharp results concerning the geometry of some Severi varieties.
A R.D.M. Accola, On Castelnuovo’s inequality for algebraic curves, I, Trans. Am. Math. Soc., 251 (1979), 357-373.
ACGH E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris, Geometry of Algebraic Curves, Springer, Berlin, 1985.
BPV W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Springer, Berlin, 1984.
Bog F. Bogomolov, Holomorphic tensors and vector bundles on projective varieties, Math. USSR Isvestija, 13 (1979), 499-555.
Cat1 F. Catanese, Fibred surfaces, varieties isogenus to a product and related moduli spaces, preprint (1999).
CC L. Chiantini and C. Ciliberto, On the Severi varieties of surfaces in $\mathbb {P}^3$, J. Alg. Geometry, 8 (1999), 67-83.
CS L. Chiantini and E. Sernesi, Nodal curves on surfaces of general type, Math. Ann., 307 (1997), 41-56.
Ciro C. Ciliberto, Alcune applicazioni di un classico procedimento di Castelnuovo, Seminari di Geometria 1982-83, Univ. di Bologna - Istituto di Geometria, (1984), 17-43.
F F. Flamini, Families of nodal curves on projective surfaces, Ph.D thesis, Consortium Universities of Rome “La Sapienza" and “Roma Tre" (1999)
F1 F. Flamini, Some results of regularity for Severi varieties of projective surfaces, Comm. Algebra 29 (2001), 2297–2311.
F2 F. Flamini and C. Madonna, Geometric linear normality for nodal curves on some projective surfaces, Boll. Un. Mat. Ital. Sez. B Artic. Ric. Mat. (8) 4 (2001), 269–283.
Fr F. Friedman, Algebraic surfaces and holomorphic vector bundles (UTX), Springer-Verlag, New York, 1998.
GH P. Griffiths and J. Harris, Residues and $0$-cycles on algebraic varieties, Ann. Math., 108 (1978), 461-505.
H J. Harris, On the Severi problem, Invent. Math., 84 (1986), 445-461.
HM J. Harris and I. Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998.
H1 R. Hartshorne, Ample Subvarieties of Algebraic Varieties, Springer LNM, 156, Springer-Verlag, Berlin, 1970.
H2 R. Hartshorne, Algebraic Geometry (GTM No. 52), Springer-Verlag, New York-Heidelberg, 1977.
Ho R. Horikawa, On deformations of holomorphic maps I, J. Math. Soc. Japan, 25 (1973), 372-396.
J K.W. Johnson, Immersion and embedding of projective varieties, Acta mathematica, 140 (1978), 49-74.
K S. L. Kleiman, The enumerative theory of singularities, in Nordic Summer School/NAVF - Symposium in Mathematics, Oslo (1976), 297-396.
Mat H. Matsumura, On algebraic groups of birational transformations, Rend. Acc. Naz. Linc., ser. 8, 34 (1963), 151-155.
Miy Y. Miyaoka, T. Peternell, Geometry of higher dimensional algebraic varieties, DMV-Seminar, Bd. 26, Birkhäuser, Basel, 1997.
Mu D. Mumford, Lectures on curves on an algebraic surface, Princeton University Press, Princeton, 1966.
OSS C. Okonek, M. Schneider and H. Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, 3, Boston-Basel-Stuttgart, Birkhäuser, 1980.
Par G. Pareschi, Components of the Hilbert scheme of smooth space curves with the expected number of moduli, Manuscripta Math., 63 (1989), 1-16.
Reid M. Reid, Bogomolov’s theorem $c_1^2 \leq 4 c_2$, Proc. Internat. Symposium on Alg. Geom., Kyoto, (1977), 633-643.
Schn M. Schneider, Symmetric differential forms as embedding obstructions and vanishing theorems, J. Alg. Geom., 1 (1992), 175-181.
Schoen C. Schoen, Varieties dominated by product varieties, International J. Math., 7 (1996), 541-571.
S E. Sernesi, On the existence of certain families of curves, Invent. Math., 75, (1984), 25-57.
Sev F. Severi, Vorlesungen über algebraische Geometrie, Teubner, Leipzig, 1921.
Shaf I.R. Shafarevich (Ed.), Algebraic Geometry II. Cohomology of algebraic varieties. Algebraic surfaces, Encyclopaedia of Mathematical Sciences, 35, Springer, Berlin, 1995.
SS B. Shiffman and A.J. Sommese, Vanishing Theorems on Complex Manifolds, Progress in Mathematics, 56, Boston-Basel-Stuttgart, Birkhäuser, 1985.
Tan S.L. Tan, Cayley-Bacharach property of an algebraic variety and Fujita’s conjecture, J. Alg. Geom., 9 (2000), 201-222.
W J.M. Wahl, Deformations of plane curves with nodes and cusps, Am. J. Math., 96 (1974), 529-577.
A R.D.M. Accola, On Castelnuovo’s inequality for algebraic curves, I, Trans. Am. Math. Soc., 251 (1979), 357-373.
ACGH E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris, Geometry of Algebraic Curves, Springer, Berlin, 1985.
BPV W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Springer, Berlin, 1984.
Bog F. Bogomolov, Holomorphic tensors and vector bundles on projective varieties, Math. USSR Isvestija, 13 (1979), 499-555.
Cat1 F. Catanese, Fibred surfaces, varieties isogenus to a product and related moduli spaces, preprint (1999).
CC L. Chiantini and C. Ciliberto, On the Severi varieties of surfaces in $\mathbb {P}^3$, J. Alg. Geometry, 8 (1999), 67-83.
CS L. Chiantini and E. Sernesi, Nodal curves on surfaces of general type, Math. Ann., 307 (1997), 41-56.
Ciro C. Ciliberto, Alcune applicazioni di un classico procedimento di Castelnuovo, Seminari di Geometria 1982-83, Univ. di Bologna - Istituto di Geometria, (1984), 17-43.
F F. Flamini, Families of nodal curves on projective surfaces, Ph.D thesis, Consortium Universities of Rome “La Sapienza" and “Roma Tre" (1999)
F1 F. Flamini, Some results of regularity for Severi varieties of projective surfaces, Comm. Algebra 29 (2001), 2297–2311.
F2 F. Flamini and C. Madonna, Geometric linear normality for nodal curves on some projective surfaces, Boll. Un. Mat. Ital. Sez. B Artic. Ric. Mat. (8) 4 (2001), 269–283.
Fr F. Friedman, Algebraic surfaces and holomorphic vector bundles (UTX), Springer-Verlag, New York, 1998.
GH P. Griffiths and J. Harris, Residues and $0$-cycles on algebraic varieties, Ann. Math., 108 (1978), 461-505.
H J. Harris, On the Severi problem, Invent. Math., 84 (1986), 445-461.
HM J. Harris and I. Morrison, Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998.
H1 R. Hartshorne, Ample Subvarieties of Algebraic Varieties, Springer LNM, 156, Springer-Verlag, Berlin, 1970.
H2 R. Hartshorne, Algebraic Geometry (GTM No. 52), Springer-Verlag, New York-Heidelberg, 1977.
Ho R. Horikawa, On deformations of holomorphic maps I, J. Math. Soc. Japan, 25 (1973), 372-396.
J K.W. Johnson, Immersion and embedding of projective varieties, Acta mathematica, 140 (1978), 49-74.
K S. L. Kleiman, The enumerative theory of singularities, in Nordic Summer School/NAVF - Symposium in Mathematics, Oslo (1976), 297-396.
Mat H. Matsumura, On algebraic groups of birational transformations, Rend. Acc. Naz. Linc., ser. 8, 34 (1963), 151-155.
Miy Y. Miyaoka, T. Peternell, Geometry of higher dimensional algebraic varieties, DMV-Seminar, Bd. 26, Birkhäuser, Basel, 1997.
Mu D. Mumford, Lectures on curves on an algebraic surface, Princeton University Press, Princeton, 1966.
OSS C. Okonek, M. Schneider and H. Spindler, Vector bundles on complex projective spaces, Progress in Mathematics, 3, Boston-Basel-Stuttgart, Birkhäuser, 1980.
Par G. Pareschi, Components of the Hilbert scheme of smooth space curves with the expected number of moduli, Manuscripta Math., 63 (1989), 1-16.
Reid M. Reid, Bogomolov’s theorem $c_1^2 \leq 4 c_2$, Proc. Internat. Symposium on Alg. Geom., Kyoto, (1977), 633-643.
Schn M. Schneider, Symmetric differential forms as embedding obstructions and vanishing theorems, J. Alg. Geom., 1 (1992), 175-181.
Schoen C. Schoen, Varieties dominated by product varieties, International J. Math., 7 (1996), 541-571.
S E. Sernesi, On the existence of certain families of curves, Invent. Math., 75, (1984), 25-57.
Sev F. Severi, Vorlesungen über algebraische Geometrie, Teubner, Leipzig, 1921.
Shaf I.R. Shafarevich (Ed.), Algebraic Geometry II. Cohomology of algebraic varieties. Algebraic surfaces, Encyclopaedia of Mathematical Sciences, 35, Springer, Berlin, 1995.
SS B. Shiffman and A.J. Sommese, Vanishing Theorems on Complex Manifolds, Progress in Mathematics, 56, Boston-Basel-Stuttgart, Birkhäuser, 1985.
Tan S.L. Tan, Cayley-Bacharach property of an algebraic variety and Fujita’s conjecture, J. Alg. Geom., 9 (2000), 201-222.
W J.M. Wahl, Deformations of plane curves with nodes and cusps, Am. J. Math., 96 (1974), 529-577.
Additional Information
F. Flamini
Affiliation:
Dipartimento di Matematica, Universita’ degli Studi di Roma - “Roma Tre", Largo San Leonardo Murialdo, 1 - 00146 Roma, Italy
MR Author ID:
650600
Email:
flamini@matrm3.mat.uniroma3.it
Received by editor(s):
July 21, 2000
Published electronically:
June 10, 2002
Additional Notes:
The author is a member of GNSAGA-INdAM
