TY - JOUR

T1 - Lines on K3 quartic surfaces in characteristic 3

AU - Veniani, Davide Cesare

N1 - Funding Information: The author acknowledges the financial support of the research training group GRK 1463 “Analysis, Geometry and String Theory”.

PY - 2022/3

Y1 - 2022/3

N2 - We investigate the number of straight lines contained in a K3 quartic surface X defined over an algebraically closed field of characteristic 3. We prove that if X contains 112 lines, then X is projectively equivalent to the Fermat quartic surface; otherwise, X contains at most 67 lines. We improve this bound to 58 if X contains a star (ie four distinct lines intersecting at a smooth point of X). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided.

AB - We investigate the number of straight lines contained in a K3 quartic surface X defined over an algebraically closed field of characteristic 3. We prove that if X contains 112 lines, then X is projectively equivalent to the Fermat quartic surface; otherwise, X contains at most 67 lines. We improve this bound to 58 if X contains a star (ie four distinct lines intersecting at a smooth point of X). Explicit equations of three 1-dimensional families of smooth quartic surfaces with 58 lines, and of a quartic surface with 8 singular points and 48 lines are provided.

UR - http://www.scopus.com/inward/record.url?scp=85100540803&partnerID=8YFLogxK

U2 - 10.1007/s00229-021-01284-9

DO - 10.1007/s00229-021-01284-9

M3 - Article

AN - SCOPUS:85100540803

VL - 167

SP - 675

EP - 701

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 3-4

ER -