# Application of Grobner bases to some rational curves

• Cho, Young-Hyun (Department of Mathematics, College of Natural Sciences, Seoul National University, Seoul 151-742) ;
• Chung, Jae-Myung (Department of Mathematics, College of Natural Sciences, Seoul National University, Seoul 151-742)
• Published : 1997.11.01

#### Abstract

Let $C_d$ be the rational curve of degree d in $P_k ^3$ given parametrically by $x_0 = u^d, X_1 = u^{d - 1}t, X_2 = ut^{d - 1}, X_3 = t^d (d \geq 4)$. Then the defining ideal of $C_d$ can be minimally generated by d polynomials $F_1, F_2, \ldots, F_d$ such that $degF_1 = 2, degF_2 = \cdots = degF_d = d - 1$ and $C_d$ is a set-theoretically complete intersection on $F_2 = X_1^{d-1} - X_2X_0^{d-2}$ for every field k of characteristic p > 0. For the proofs we will use the notion of Grobner basis.

#### References

1. J. Algebra v.86 liasion, arithmetical Buchsbaum curves and monomial curves in $P^3$ H. Bresinsky;P. Schenzel;W. Vogel
2. G.T.M. v.150 Commutative algevra with a view toward algebraix geometry D. Eisenbud
3. Amer. J. Math. v.101 Complete intersection in characteristic p>0 R. Hartshorne
4. Man. Math. v.3 Generators and relations of abelian semogroups and semigroup ring J. Herzog
5. Lecture Notes in Mathematics v.997 Some curves in $P^3$ are set-theorti complete intresections R. Lobbiano;G. Valla
6. Commutative Algebra v.2 O. Zariski;P. Samuel