2010/01/07 by Francisco Javier Gallego, F. J. Gallego, Gallego, F. J. +3
Computer Science · Mathematics · #14J10 #14J29 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:14J10 #msc:14J29
paper · pdf · doi:10.48550/arxiv.1001.1128
20 pages
arxiv created 2010/01/07 · openalex publication_date 2010/01/07 · arxiv updated 2010/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
In this article we study the bicanonical map ϕ2 of quadruple Galois canonical covers X of surfaces of minimal degree. We show that ϕ2 has diverse behavior and exhibit most of the complexities that are possible for a bicanonical map of surfaces of general type, depending on the type of X. There are cases in which ϕ2 is an embedding, and if so happens, ϕ2 embeds X as a projectively normal variety, and cases in which ϕ2 is not an embedding. If the latter, ϕ2 is finite of degree 1, 2 or 4. We also study the canonical ring of X, proving that it is generated in degree less than or equal to 3 and finding the number of generators in each degree. For generators of degree 2 we find a nice general formula which holds for canonical covers of arbitrary degrees. We show that this formula depends only on the geometric and the arithmetic genus of X.