Given a complex $K3$ surface $S$ in a projective space $\mathbb{P}^N$, $N>3$, is there some cohomological characterization that allows us to see if $S$ contains some conic $C$? I am particularly interested in the case $N=6$.
Here we take a conic as a quadric curve contained in a plane.
Take $L$ the polarization on $S$ associated to the embedding $S\hookrightarrow\mathbb{P}^N$ (there is no hyperplane containing $S$); the couple $(S,L)$ is uniquely determined by the even unimodular lattice $(H^2(S,\mathbb{Z}),(\cdot,\cdot))$, where $(\cdot,\cdot)$ is the intersection form. So I think that we should be able to keep trace, in $NS(S)\subset H^2(S,\mathbb{Z})$ or at least in $H^2(S,\mathbb{Z})$, of a conic $C\subset S$ .
As $C$ is a quadric, I thought that there would be some class $\eta\in NS(S)$ such that $(L,\eta)=2$ and either $(\eta,\eta)=0$ or $(\eta,\eta)=-2$. However I am confused because, since $C$ come from a linear section of $S$, its class in $NS(S)$ should come from $L$, but the general linear section of $S$ is irreducible. I think that the presence of a conic should give some divisorial condition in the moduli space of polarized $K3$ surfaces.
I know my guesses are a bit confused, but I tried to sum up my toughts. I referred to "Moduli of K3 Surfaces and Irreducible Symplectic Manifolds" by Gritsenko, Hulek and Sankaran, and "Lectures on K3 surfaces", Huybrechts, for the correspondence between $K3$'s and second cohomology.