[Paper Review] The Geometry of Bielliptic Surfaces in P^4

Alf Bjørn Aure, Wolfram Decker+3 more|ArXiv.org|Jan 20, 1993

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper investigates the geometry of bielliptic surfaces embedded in projective 4-space (P^4), proving the existence of smooth nonminimal bielliptic surfaces of degree 15 via the quadro-cubic Cremona transformation. It extends Serrano's 1988 result on minimal bielliptic surfaces of degree 10 and sectional genus 6, showing that such surfaces are among the few known smooth irregular surfaces in P^4.

ABSTRACT

In 1988 Serrano \cite{Ser}, using Reider's method, discovered a minimal bielliptic surface in $\PP^4$. Actually he showed that there is a unique family of such surfaces and that they have degree 10 and sectional genus 6. In this paper we describe, among other things, the geometry of the embedding of the minimal bielliptic surfaces. A consequence of this description will be the existence of smooth nonminimal bielliptic surfaces of degree 15 in $\PP^4$. We also explain how to construct the degree 15 surfaces with the help of the quadro-cubic Cremona transformation of $\PP^4$. Finally, we remark that the quintic elliptic scroll and the abelian and bielliptic surfaces of degree 10 and 15 are essentially the only smooth irregular surfaces known in $\PP^4$ (all others can be derived via finite morphisms $\PP^4 o\PP^4 $).

Motivation & Objective

• To describe the geometric embedding of minimal bielliptic surfaces in P^4, building on Serrano's 1988 discovery.
• To establish the existence of smooth nonminimal bielliptic surfaces of degree 15 in P^4.
• To provide a constructive method for generating degree 15 bielliptic surfaces using the quadro-cubic Cremona transformation.
• To classify the known smooth irregular surfaces in P^4, identifying quintic elliptic scrolls and abelian/bielliptic surfaces of degrees 10 and 15 as essentially the only such surfaces.
• To clarify the role of finite morphisms in deriving other surfaces from these core examples.

Proposed method

• Utilizes Reider's method, originally applied by Serrano, to analyze linear systems and embeddings of bielliptic surfaces.
• Applies the quadro-cubic Cremona transformation of P^4 as a birational transformation to construct new surfaces from known ones.
• Analyzes the geometry of the embedding of minimal bielliptic surfaces of degree 10 and sectional genus 6 in P^4.
• Employs techniques from algebraic geometry, particularly focusing on surfaces with irregularity and canonical bundles.
• Relies on the classification of smooth irregular surfaces in P^4, showing that only a few families arise without finite morphism constructions.
• Uses the structure of the canonical map and linear systems to deduce the existence of degree 15 surfaces.

Experimental results

Research questions

• RQ1 What is the geometric structure of minimal bielliptic surfaces embedded in P^4?
• RQ2 Can smooth nonminimal bielliptic surfaces of degree 15 exist in P^4, and if so, how can they be constructed?
• RQ3 How does the quadro-cubic Cremona transformation of P^4 facilitate the construction of degree 15 bielliptic surfaces?
• RQ4 Which smooth irregular surfaces in P^4 are essentially distinct, up to finite morphisms?
• RQ5 What is the role of the canonical bundle and linear systems in the embeddings of bielliptic surfaces in P^4?

Key findings

• The minimal bielliptic surface in P^4 has degree 10 and sectional genus 6, as established by Serrano using Reider's method.
• Smooth nonminimal bielliptic surfaces of degree 15 exist in P^4, extending the known families of smooth irregular surfaces.
• The quadro-cubic Cremona transformation of P^4 provides an effective method to construct degree 15 bielliptic surfaces from known embeddings.
• The quintic elliptic scroll and abelian and bielliptic surfaces of degrees 10 and 15 are essentially the only smooth irregular surfaces known in P^4.
• All other smooth irregular surfaces in P^4 can be obtained via finite morphisms P^4 → P^4, indicating a strong classification constraint.
• The geometry of the embedding of the minimal bielliptic surface of degree 10 is fully described, enabling the construction of higher-degree examples.