Title: Smooth surfaces with maximally many lines

Abstract: How many lines can lie on a smooth surface of degree d?  In 1849, Cayley and Salmon proved their famous result that every smooth projective surface of degree 3 contains 27 lines.  In higher degrees, we know that not all surfaces of degree d have the same number of lines (many have no lines at all), but bounding the maximal number of lines on a smooth complex surface of degree d is a classical subject with a history dating back at least to Clebsch’s 1861 bound of d(11d-24), with the best known upper bound  of 11d^2 - 30d + 18 due to Bauer and Rams in 2023.  However, over fields of positive characteristic, there are smooth surfaces which contain more lines than these upper bounds.  In this talk, I’ll give a new upper bound that holds over any field without assumptions on the characteristic.  In addition, we’ll fully classify those surfaces of degree d which attain our upper bound.  This talk is based on joint work with Tim Ryan and Karen Smith.