NYJM Logo

New York Journal of Mathematics
Volume 27 (2021), 1305-1327

  

Stephen McKean, Daniel Minahan, and Tianyi Zhang

All lines on a smooth cubic surface in terms of three skew lines

view    print


Published: September 3, 2021.
Keywords: Cubic surface, real lines, enumerative geometry.
Subject: 14N15.

Abstract
Jordan showed that the incidence variety of a smooth cubic surface containing 27 lines has solvable Galois group over the incidence variety of a smooth cubic surface containing 3 skew lines. As noted by Harris, it follows that for any smooth cubic surface, there exist formulas for all 27 lines in terms of any 3 skew lines. In response to a question of Farb, we compute these formulas explicitly. We also discuss how these formulas relate to Schläfli's count of lines on real smooth cubic surfaces.

Acknowledgements

We thank Benson Farb for asking this paper's motivating question. We also thank Matt Baker, Dan Margalit, Joe Rabinoff, Bernd Sturmfels, and Jesse Wolfson for helpful suggestions and support. We thank the anonymous referee for their detailed comments and suggestions that greatly improved the clarity and accuracy of the paper. Finally, we are especially grateful to Steve Trettel for the included graphics.


Author information

Stephen McKean:
Department of Mathematics
Duke University
Durham, NC 27708, USA

mckean@math.duke.edu

Daniel Minahan:
School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332, USA

dminahan6@gatech.edu

Tianyi Zhang:
School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332, USA

kafuka@gatech.edu