## Osaka Journal of Mathematics

### Curves in projective spaces and their index of regularity

Edoardo Ballico

#### Abstract

For all integers $n \ge 3$ we show the existence of many triples $(d,g,\rho)$ such that there is a smooth non-degenerate curve $C \subset \mathbf{P}^n$ with degree $d$, genus $g$ and index of regularity $\rho$. The curve $C$ lies in a smooth $K3$ surface $S \subset \mathbf{P}^n$.

#### Article information

Source
Osaka J. Math., Volume 43, Number 1 (2006), 179-181.

Dates
First available in Project Euclid: 28 April 2006

https://projecteuclid.org/euclid.ojm/1146243000

Mathematical Reviews number (MathSciNet)
MR2222407

Zentralblatt MATH identifier
1100.14024

Subjects
Primary: 14H50: Plane and space curves
Secondary: 14N50

#### Citation

Ballico, Edoardo. Curves in projective spaces and their index of regularity. Osaka J. Math. 43 (2006), no. 1, 179--181. https://projecteuclid.org/euclid.ojm/1146243000

#### References

• E. Ballico, N. Chiarli and S. Greco: On the existence of $k$-normal curves of given degree and genus in projective spaces, Collect. Math. 55 (2004), 269–277.
• L. Gruson, R. Lazarsfeld and C. Peskine: On a theorem of Castenuovo, and the equations defining space curves, Invent. Math. 72 (1983), 491–506.
• A.L. Knutsen: Smooth curves on projective $K3$ surfaces, Math. Scand. 90 (2000), 215–231.
• S. Mori: On degree and genera of curves on smooth quartic surfaces in $\mathbf{P}^3$, Nagoya Math. J. 96 (1984), 127–132.
• D. Mumford: Varieties defined by quadratic equations; in Questions on Algebraic Varieties, Cremonese, Rome, 1970, 30–100.