Algebraic Geometry Seminar
Tim Ryan
North Dakota State University
Surfaces with maximally many lines
Abstract: While the general surface of degree at least 4 in projective 3-space contains no lines, the maximum possible number of lines on any surface of degree at least 4 over a field k is a classical question dating back to at least Clebsch's work in 1861. When the degree is less than the (positive) characteristic and always in characteristic 0, the number of lines has an upper bound which is quadratic in the degree when the degree is at least 4. In contrast, once the degree is at least one more than the characteristic, it has long been known that there are surfaces with vastly more lines. In this talk, we answer this classical question over an arbitrary field. In particular, we prove that the maximum number of lines on any smooth surface of degree d over any field k is $d^4-3d^3+3d^2$ and show that, up to projective equivalence, a unique surface obtains this sharp upper bound in the infinitely many degrees and characteristics where it is obtained.
Monday April 14, 2025 at 3:00 PM in 636 SEO