The fact that a compact complex 2-dimensional manifold \(X_d\) given as the set of zeroes of a degree-d homogeneous polynomial in 3-dimensional projective space contains at most \(d(11d-24)\) lines was shown already by G. Salmon around 1840. Salmon, Cayley and Clebsch were also able to prove that the above claim is optimal for every cubic surface (i.e. for \(d=3\)). It is a far more difficult question whether the above bound is sharp (resp. what is the maximum number of lines on \(X_d\)'s) once we fix a degree \(d\) that exceeds 3. After a brilliant (but based on false claims on configurations of lines) argument by B.Segre (1943), the first correct proof of the claim that a smooth quartic surface contains at most 64 lines was given in 2012 (M. Schuett-S.R.). Hardly anything is known on number and configurations of lines on degree-d surfaces when d is at least 5.

In my talk I will discuss the classical argument by Segre and sketch the proof of the sharp bound for \(d=4\). I will explain why some claims by Segre are false by providing a detailed picture of the geometry of quartics \(X_4\) with so-called lines of the second kind. Finally, I will explain what happens if we consider degree 4 surfaces in 3-dimensional affine space, allow the considered surface not to be a complex manifold in a finite set of points (resp. along a curve), replace the field of complex numbers with an algebraically closed field of positive characteristic. If time permits, I will state some results for degree 5 (based on joint work with Prof. M. Schuett (LUH Hannover)).

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