Smooth Curves on Projective $K3$ Surfaces

Andreas Leopold Knutsen

Abstract


In this paper we give for all $n \geq 2$, $d>0$, $g \geq 0$ necessary and sufficient conditions for the existence of a pair $(X,C)$, where $X$ is a $K3$ surface of degree $2n$ in $\mathrm{P}^{n+1}$ and $C$ is a smooth (reduced and irreducible) curve of degree $d$ and genus $g$ on $X$. The surfaces constructed have Picard group of minimal rank possible (being either $1$ or $2$), and in each case we specify a set of generators. For $n \geq 4$ we also determine when $X$ can be chosen to be an intersection of quadrics (in all other cases $X$ has to be an intersection of both quadrics and cubics). Finally, we give necessary and sufficient conditions for $\mathcal O_C (k)$ to be non-special, for any integer $k \geq 1$.

Full Text:

PDF


DOI: http://dx.doi.org/10.7146/math.scand.a-14371

Refbacks

  • There are currently no refbacks.
This website uses cookies to allow us to see how the site is used. The cookies cannot identify you or any content at your own computer.
OK


ISSN 0025-5521 (print) ISSN 1903-1807 (online)

Hosted by the Royal Danish Library