On the curves of contact on surfaces in a projective space. III
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Abstract:
Suppose a smooth curve $C$ is a set-theoretic complete intersection of two surfaces $F$ and $G$ with the multiplicity of $F$ along $C$ less than or equal to the multiplicity of $G$ along $C$. One obtains a relation between the degrees of $C$, $F$ and $G$, the genus of $C$, and the multiplicity of $F$ along $C$ in case $F$ has only ordinary singularities. One obtains (in the characteristic zero case) that a nonsingular rational curve of degree 4 in $\mathbf {P}^3$ is not set-theoretically an intersection of 2 surfaces, provided one of them has at most ordinary singularities. The same result holds for a general nonsingular rational curve of degree $\geq 5$.References
- C. Bănică and O. Forster, Multiplicity structures on space curves, The Lefschetz centennial conference, Part I (Mexico City, 1984) Contemp. Math., vol. 58, Amer. Math. Soc., Providence, RI, 1986, pp. 47–64. MR 860403, DOI 10.1090/conm/058.1/860403
- M. Boratyński, On the curves of contact on surfaces in a projective space, Algebraic $K$-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989) Contemp. Math., vol. 126, Amer. Math. Soc., Providence, RI, 1992, pp. 1–8. MR 1156498, DOI 10.1090/conm/126/00505
- M. Boratyński, On the curves of contact on surfaces in a projective space. II, Rend. Sem. Mat. Univ. Politec. Torino 48 (1990), no. 4, 439–455 (1993). Commutative algebra and algebraic geometry, I (Italian) (Turin, 1990). MR 1217771
- M. Boratyński, Locally complete intersection multiple structures on smooth algebraic curves, Proc. Amer. Math. Soc. 115 (1992), no. 4, 877–879. MR 1120504, DOI 10.1090/S0002-9939-1992-1120504-1
- Hartmut Lindel, On projective modules over polynomial rings over regular rings, Algebraic $K$-theory, Part I (Oberwolfach, 1980) Lecture Notes in Math., vol. 966, Springer, Berlin-New York, 1982, pp. 169–179. MR 689374
- Lorenzo Robbiano, A problem of complete intersections, Nagoya Math. J. 52 (1973), 129–132. MR 332807
- Paolo Valabrega and Giuseppe Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93–101. MR 514892
Additional Information
- M. Boratynski
- Affiliation: Dipartimento di Matematica, via E Orabona 4, 70125 Bari, Italy
- Email: boratyn@pascal.dm.uniba.it
- Received by editor(s): July 25, 1994
- Received by editor(s) in revised form: December 13, 1994, and June 27, 1995
- Additional Notes: This research was supported by the funds of the Italian Ministry of Education and Scientific Research (MURST)
- Communicated by: Wolmer V. Vasconcelos
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 329-338
- MSC (1991): Primary 14H50
- DOI: https://doi.org/10.1090/S0002-9939-97-03532-6
- MathSciNet review: 1346964