# Lines on smooth polarized $K3$-surfaces

## TL;DR

This paper establishes precise maximum counts for lines on smooth polarized K3 surfaces in projective spaces, confirming conjectured bounds for specific degrees and dimensions.

## Contribution

It provides sharp bounds on the number of lines on smooth polarized K3 surfaces for each degree, confirming previous conjectures in key cases.

## Key findings

- Maximum of 42 lines on sextic K3 surfaces in P^4
- Maximum of 36 lines on octic K3 surfaces in P^5
- Confirmed conjectured bounds for these cases

## Abstract

For each integer $D\ge3$, we give a sharp bound on the number of lines contained in a smooth complex $2D$-polarized $K3$-surface in $\mathbb{P}^{D+1}$. In the two most interesting cases of sextics in $\mathbb{P}^4$ and octics in $\mathbb{P}^5$, the bounds are $42$ and $36$, respectively, as conjectured in an earlier paper.