[Paper Review] Severi varieties and self rational maps of K3 surfaces

Thomas Dedieu|arXiv (Cornell University)|Apr 24, 2007

Algebraic Geometry and Number Theory11 references4 citations

TL;DR

This paper investigates dominant self-rational maps on complex projective K3 surfaces with Picard group ℤ, focusing on the interplay between algebraic and topological degrees, ramification, and exceptional divisors. It proves that for degree 4 maps, the number of blown-up points is bounded by 24 if the exceptional divisor has depth 1, or exactly one point if depth is 2, providing strong numerical constraints on such maps.

ABSTRACT

Self-rational maps of generic algebraic K3 surfaces are conjectured to be trivial. We relate this conjecture to a conjecture concerning the irreducibility of the universal Severi varieties parametrizing nodal curves of given genus and degree lying on some K3 surface. We also establish a number of numerical constraints satisfied by such non trivial rational maps, that is of topological degree >1.

Motivation & Objective

• To investigate the existence and geometric properties of dominant self-rational maps on K3 surfaces with Picard group ℤ.
• To analyze the topological and algebraic degrees of such maps and their implications for the surface's geometry.
• To constrain the number and configuration of exceptional curves arising from elimination of indeterminacies.
• To test the validity of Conjecture 1, which posits no such maps exist for generic K3 surfaces with degree >1.
• To provide numerical obstructions using Chern classes and ramification data to rule out or limit possible self-rational maps.

Proposed method

• Uses elimination of indeterminacies via blow-ups to resolve the rational map φ:S⇢S into a morphism φ̃:S̃→S.
• Applies the holomorphic dynamical framework to relate the map's degree and ramification divisor R to the Chern classes of the cotangent bundle.
• Employs exact sequences involving Ω_S̃ and the pullback of Ω_S to compute c₂(Ω_S̃) in terms of deg(φ) and contributions from exceptional divisors.
• Analyzes the restriction of the differential dφ̃ to exceptional curves E_i to bound the degree of the line bundle L_i on each E_i.
• Uses the conormal exact sequence and the ramification condition to derive deg(K_i) ≤ -2 and deg(L_i) ≥ 1, leading to c₂(L_i) ≤ -2.
• Applies the formula c₂(Ω_S̃) = deg(φ)·c₂(Ω_S) + Σc₂(L_i) and the known value c₂(Ω_S̃) = 24 + p to derive bounds on p.

Experimental results

Research questions

• RQ1 Can a K3 surface with Picard group ℤ admit a dominant self-rational map of degree l > 1?
• RQ2 What constraints does the structure of the exceptional divisor impose on the number and configuration of blown-up points?
• RQ3 How do the algebraic and topological degrees of such maps relate in the context of K3 surfaces?
• RQ4 What numerical obstructions arise from Chern class computations on the blow-up surface?
• RQ5 Are there any K3 surfaces with Picard group ℤ that support self-rational maps of degree 4 or 9, and what are the possible values of l?

Key findings

• For a self-rational map φ of degree 4 on a K3 surface with Picard group ℤ, if the exceptional divisor has depth 1, then the number of blown-up points p satisfies p ≤ 24.
• If the exceptional divisor has depth 2, then only a single point can be blown up, regardless of the degree.
• The bound p ≤ 8(deg(φ) - 1) holds when the exceptional curves are disjoint and the map has depth 1.
• For degree 4 maps, the possible values of l (the algebraic degree) are restricted: l = 6,8,10,… for genus 2, and l = 6,10,14,… for genera 3,4,5.
• For degree 9 maps, the minimal possible l values are l = 5,7,9,… for genera 2 and 3, and l = 9,15,21,… for genus 4, with l = 5,11,13,19,… for genus 5.
• The results show that the hypotheses of Proposition 3.6 are always satisfied when deg(φ) = 4, indicating strong numerical constraints on such maps.