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On the existence of certain families of curves. (English) Zbl 0541.14024
Let \(V\) denote an open irreducible subset of the Hilbert-scheme of \(\mathbb P^r\) parametrizing irreducible and smooth (if \(r\geq 3)\) resp. nodal (if \(r=2)\) curves of genus \(g\) and \(\pi\colon V\to \mathcal M_ g\) the natural map into the moduli space of curves of genus \(g\). \(V\) is called to have the expected number of moduli if \(\dim \pi(V)=\min(3 g-3, 3g - 3+\rho(g,r,n))\), where \(\rho(g,r,n)\) is the Brill-Noether number. It is the aim of the paper to construct such families of curves. For plane curves the result is quite complete. It is shown that for all \(g\) and \(n\geq 5\) such that \(n-2\leq g\leq \binom{n-1}{2}\) there is an irreducible component of the family of plane irreducible nodal curves of degree \(n\) and genus \(g\), having the expected number of moduli. In the range \(n-2\leq g\leq 3n/2-3\) (resp. \(2n-4\leq g\leq \binom{n-1}{2}\) this result was known (resp. independently proved) by P. Griffiths and J. Harris [Duke Math. J. 47, 233–272 (1980; Zbl 0446.14011)] (resp. by Coppens).
For \(r\geq 3\) the result is: for all \(g\) and \(n\geq r+1\) such that \(n-r\leq g\leq(r(n-r)-1)/(r-1)\) (resp. \(n-3\leq g\leq 3n-18\) if \(r=3)\) there is an open set of an irreducible component of the Hilbert scheme of \(\mathbb P^r\) parametrizing smooth irreducible curves of degree \(n\) and genus \(g\), which has the expected number of moduli. An immediate consequence is a special case of a theorem of D. Eisenbud and J. Harris [Invent. Math. 74, 371–418 (1983; Zbl 0527.14022)] on the existence of very ample line bundles of a prescribed type.
The proofs of the theorems are given by induction, the induction step being roughly as follows: Start with a particular curve \(C\) in \(\mathbb P^n\) whose existence is known, construct a new curve by adding a particular rational curve \(\gamma\) and get a new curve \(C'\) by flat smoothing of \(C\cup \gamma\).

14H10 Families, moduli of curves (algebraic)
Full Text: DOI EuDML
[1] Arbarello, E., Cornalba, M.: Su una propriet√† notevole dei morfismi di una curva a moduli generali in uno spazio proiettivo. Rend. Sem. Mat. Univers. Politecn. Torino38, 87-99 (1980) · Zbl 0478.14016
[2] Arbarello, E., Cornalba, M.: Su una congettura di Petri. Comment. Math. Helvetici56, 1-38 (1981) · Zbl 0505.14002
[3] Coppens, M.R.M.: Plane models of smooth curves over ?. Preprint, 1982
[4] Eisenbud, D., Harris, J.: Divisors on general curves and cuspidal rational curves. Preprint · Zbl 0527.14022
[5] Fulton, W., Lazarsfeld, R.: On the connectedness of degeneracy loci and special divisors. Acta Math.146, 271-283 (1981) · Zbl 0469.14018
[6] Gieseker, D.: Stable curves and special divisors: Petri’s conjecture. Invent. Math.66, 251-275 (1982) · Zbl 0522.14015
[7] Gieseker, D.: A construction for special space curves. Preprint · Zbl 0558.14024
[8] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York, Chichester, Brisbane, Toronto: John Wiley & Sons, 1978 · Zbl 0408.14001
[9] Griffiths, P., Harris, J.: On the variety of special linear systems on a general algebraic curve. Duke Math. J.47, 233-272 (1980) · Zbl 0446.14011
[10] Gruson, L., Peskine, C.: Genre des courbes de l’espace projectif. II · Zbl 0517.14007
[11] Grothendieck, A.: Les schemas the Hilbert. Sem. Bourbaki,221 (1960-61)
[12] Hartshorne, R.: Algebraic Geometry. Berlin-Heidelberg-New York: Springer 1977 · Zbl 0367.14001
[13] Horikawa, E.: On deformations of holomorphic maps, I. J. Math. Soc. Japan25, 372-396 (1973) · Zbl 0254.32022
[14] Kleiman, S.L., Laksov, D.: On the existence of special divisors. Amer. J. Math.94, 431-436 (1972) · Zbl 0251.14005
[15] Laudal, O.A.: Formal moduli of algebraic structures. Lecture Notes in Mathematics, vol. 754. Berlin-Heidelberg-New York: Springer 1979 · Zbl 0438.14007
[16] Lichtenbaum, S., Schlessinger, M.: The cotangent complex of a morphism. Trans. A.M.S.128, 41-70 (1967) · Zbl 0156.27201
[17] Mumford, D.: Lectures on curves on an algebraic surface. Princeton: Princeton University Press 1966 · Zbl 0187.42701
[18] Sacchiero, G.: Fibrati normali di curve razionali dello spazio proiettivo. Ann. Univ. Ferrara sez. VII, Sc. Mat.27, 33-40 (1980) · Zbl 0496.14020
[19] Schlessinger, M.: Functors of Artin rings. Trans. A.M.S.130, 208-222 (1968) · Zbl 0167.49503
[20] Saint Donat, B.: On Petri’s analysis of the linear system of quadrics through a canonical curve. Math. Ann.206, 157-175 (1973) · Zbl 0315.14010
[21] Sernesi, E.: Small deformations of global complete intersections. Boll. U.M.I. (4)12, 138-146 (1975) · Zbl 0333.32019
[22] Severi, F.: Vorlesungen uber algebraische Geometrie. Leipzig: 1921 · JFM 48.0687.01
[23] Wahl, J.: Deformations of plane curves with nodes and cusps. Amer. J. Math.96, 529-577 (1974) · Zbl 0299.14008
[24] Wirtinger, W.: Ein anderer Beweis des Satzes, dass zwei singularitaten freie Kurven von einer Ordnungn?4 notwendig kollinear sind, wenn sie eindeutig aufeinander bezogen sind. Sitzungsb. d. Math. Naturw. Abt.III, 145-147 (1932) · Zbl 0006.17904
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