turkmath.org

Counting or estimating the number of lines or, more generally, low degree rational curves on a polarized algebraic surface is a classical problem going back almost 1.5 centuries. After a brief historical excurse, I will try to give an account of the considerable progress made in the subject in the last decade or so, mainly related to various(quasi-)polarizations of $K3$-surfaces:

$\bullet$ lines on $K3$-surfaces with any polarization, 
$\bullet$ lines on low degree $K3$-surfaces with singularities, 
$\bullet$ conics on low degree $K3$-surfaces.

If time permits, I will briefly discuss other surfaces/varieties as well.

Some parts of this work are joint projects (some still in progress) with Ilia Itenberg, Slavomir Rams, Ali Sinan Sertöz.