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.

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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