Strata of smooth space curves having unstable normal bundle

Ramella, Luciana

Bollettino della Unione matematica italiana, Série 8, 2B (1999) no. 3, pp. 499-516.

Résumé

Per $d \gg g$, vengono trovate curve liscie in $\mathbb{P}^{3}$ di grado $d$ e genere $g$ aventi fibrato normale instabile con grado di instabilità $\sigma$, per ogni $1\leq\sigma\leq d-4$. Inoltre per $4g-2\leq\sigma\leq d-4$, viene trovata una famiglia di curve in $\mathbb{P}^{3}$ di grado $d$ e genere $g$ avente fibrato normale instabile con grado di instabilità $\sigma$ e formante uno strato dello schema di Hilbert della giusta dimensione che è $4d-g+1-2\sigma$.

Bibliographie

[1] E. Arbarello - M. Cornalba - P. A. Griffiths - J. Harris , Geometry of Algebraic Curves, Springer-Verlag, New York-Berlin-Heidelberg-Tokio (1985). | MR | Zbl

[2] E. Ballico - Ph. Ellia , Some more examples of curves in $\mathbb{P}^{3}$ with stable normal bundle, J. Reine Angew. Math., 350 (1984), 87-93. | MR | Zbl

[3] A. Biancofiore - E. L. Livorni , On the genus of a hyperplane section of a geometrically ruled surface, Ann. Mat. Pura Appl., 147 (1987), 173-185. | MR | Zbl

[4] N. Bourbaki , Élements de mathématiques - Algèbre commutative, f. XXVIII, ch. 1 et 2, Herman, Paris (1961). | Zbl

[5] P. Deligne - M. Rapoport , Les schémas de modules de courbes elliptiques, Lecture Notes in Math., n. 349, Springer-Verlag, Berlin, Heidelberg, New York (1973), 143-316. | MR | Zbl

[6] D. Eisenbud - A. Van De Ven , On the normal bundles of smooth rational space curves, Math. Ann., 256 (1981), 453-463. | MR | Zbl

[7] D. Eisenbud - A. Van De Ven , On the variety of smooth rational space curves with given degree and normal bundle, Invent. Math., 67 (1982), 89-100. | MR | Zbl

[8] Ph. Ellia , Exemples de courbes de $\mathbb{P}^{3}$ à fibré normal semi-stable, stable, Math. Ann., 264 (1983), 389-396. | MR | Zbl

[9] Ph. Ellia , Double structures and normal bundle of spaces curves, preprint. | MR | Zbl

[10] G. Ellingsrud - A. Hirschowitz , Sur le fibré normal des courbes gauches, C. R. Acad. Sc. Paris, 299 (1984), 245-248. | MR | Zbl

[11] G. Ellingsrud - D. Laksov , The normal bundle of elliptic space curves of degree 5, 18th Scandinavian Congress of Mathematicians, Proceedings 1980, PM 11, Birkhäuser, Boston, Basel, Stuttgart (1981). | MR | Zbl

[12] F. Ghione - G. Sacchiero , Normal bundle of rational curves in $\mathbb{P}^{3}$, Manuscripta Math., 33 (1980), 111-128. | MR | Zbl

[13] A. Grothendieck , Techniques de descente et théorèmes d'existence en géométrie algébrique-II: Le théorème d'existence et théorie formelle des modules, Sém. Bourbaki, exp. 195, 12 (1959/60). | fulltext mini-dml | Zbl

[14] A. Grothendieck , Techniques de construction et théorèmes d'existence en géométrie algébrique: Les schémas de Hilbert, Sém. Bourbaki, exp. 221, 13 (1960/61). | fulltext mini-dml | Zbl

[15] A. Grothendieck - J. Dieudonné , Éléments de géométrie algébrique. Étude locale des schémas et des morphismes des schémas, Publ. Math. IHES, 20 (1964). | fulltext mini-dml | MR | Zbl

[16] A. Grothendieck - J. Dieudonné , Éléments de géométrie algébrique. Étude locale des schémas et des morphismes des schémas, Publ. Math. IHES, 32 (1967). | fulltext mini-dml | Zbl

[17] R. Hartshorne , Algebraic Geometry, Springer-Verlag, New York, Heidelberg, Berlin (1977). | MR | Zbl

[18] A. Hirschowitz , Rank techniques and jump statifications, in Vector Bundles and Algebraic Varieties, Proc. Bombay 1984, Oxford Univ. Press (1987), 159-205. | MR | Zbl

[19] K. Hulek , The normal bundle of a curve on a quadric, Math. Ann., 258 (1981), 201-206. | MR | Zbl

[20] K. Hulek - G. Sacchiero , On the normal bundle of elliptic space curves, Arch. Math., 40 (1983), 61-68. | MR | Zbl

[21] H. Lange - M. S. Narasimhan , Maximal subbundles of rank two vector bundles on curves, Math. Ann., 266 (1983), 55-72. | MR | Zbl

[22] M. Nagata , On self-intersection number of a section on a ruled surface, Nagoya Math. J., 37 (1970), 191-196. | fulltext mini-dml | MR | Zbl

[23] P. E. Newstead , A space curve whose normal bundle is stable, J. London Math. Soc., 28 (1983), 428-434. | MR | Zbl

[24] D. Perrin , Courbes passant par $m$ points généraux de $\mathbb{P}^{3}$, Mémoires de la Soc. Math. de France, no 28/29, Suppl. au Bull. de la S.M.F., 115 (1987). | fulltext mini-dml | Zbl

[25] R. Piene , Numerical characters of a curve in projective $n$-space, in Real and Complex Singularities, Oslo 1976, Sijthoff en Noordhoff (1978), 475-495. | MR | Zbl

[26] H. Popp , Moduli Theory and Classification Theory of Algebraic Varieties, Lecture Notes in Math., n. 620, Springer-Verlag, Berlin, Heidelberg, New York (1977). | MR | Zbl

[27] G. Sacchiero , On the varieties parametrising rational space curves with fixed normal bundle, Manuscripta Math., 37 (1982), 217-228. | MR | Zbl

[28] G. Sacchiero , Exemple de courbes de $\mathbb{P}^{3}$ de fibré normal stable, Commun. Algebra, 11 (1983), 2115-2121. | MR | Zbl

[29] A. Van De Ven , Le fibré normal d'une courbe dans $\mathbb{P}^{3}$ ne se decompose pas toujours, C.R. Acad. Sc. Paris, 289 (1979), 111-113. | MR | Zbl

[30] O. Zariski , Algebraic Surfaces, 2nd ed., Springer, Berlin, Heidelberg, New York (1971). | MR | Zbl