[Paper Review] A question of Doctor Malte Wandel on automorphisms of the punctural Hilbert schemes of K3 surfaces

Keiji Oguiso|arXiv (Cornell University)|Feb 18, 2014

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper provides a positive answer to a question by Malte Wandel by constructing K3 surfaces S with finite automorphism group Aut(S) yet infinite automorphism group Aut(S^{[2]}) on their punctual Hilbert schemes of length two. The construction relies on K3 surfaces with Néron-Severi lattice isomorphic to the even hyperbolic lattice Λ of discriminant 17, which admit two distinct embeddings into P^3, inducing two distinct Beauville involutions on S^{[2]}. The key result is that S^{[2]} is not a Mori dream space, while the target S^{(2)} of the Hilbert-Chow morphism is.

ABSTRACT

We present a sufficient condition for the punctural Hilbert scheme of length two of a K3 surface with finite automorphism group to have automorphism group of infinite order in geometric terms (Theorem 2.1). We then give concrete examples (Theorem 1.2). We also discuss about Mori dream space (MDS) structures under an extermal crepant resolution (Theorems 1.2, 5.2, 4.1) from the viewpoint of automorphisms. These affirmatively answer a question of Doctor Malte Wandel.

Motivation & Objective

• To resolve a question posed by Malte Wandel regarding whether the punctual Hilbert scheme S^{[2]} of a K3 surface S can have infinite automorphism group when S itself has finite automorphism group.
• To establish a geometric criterion (Theorem 2.1) for when Aut(S^{[2]}) is infinite, based on the existence of multiple embeddings of S into projective space.
• To investigate the Mori dream space (MDS) property under extremal crepant resolutions, particularly for the Hilbert-Chow morphism μ:S^{[2]}→S^{(2)}.
• To demonstrate that S^{[2]} is not a Mori dream space while S^{(2)} is, despite the resolution being crepant and extremal.
• To provide explicit examples of K3 surfaces S with NS(S) ≅ Λ where Aut(S) is finite but Aut(S^{[2]}) is infinite, affirming the existence of such examples.

Proposed method

• Constructing K3 surfaces S with Néron-Severi lattice isomorphic to the even hyperbolic lattice Λ of discriminant 17, which ensures Picard number ρ(S) = 2.
• Using the fact that such K3 surfaces admit at least two distinct embeddings into P^3 as smooth quadric surfaces without lines, leading to two different linear systems.
• Leveraging Beauville’s involutions on S^{[2]} induced by these embeddings, which generate infinite subgroups of Aut(S^{[2]}) when the embeddings are non-isomorphic.
• Applying Kleiman’s criterion and properties of the nef cone to show that if Aut(S^{[2]}) were infinite, then the nef cone could not be a finite rational polyhedral cone.
• Using the fact that the Hilbert-Chow morphism μ:S^{[2]}→S^{(2)} is an extremal crepant resolution, and analyzing the MDS property via the action of the symmetric group Σ_n on S^n.
• Applying results from Okawa and others on the MDS property of quotients: if S^n is a MDS (which it is under finite Aut(S_k)), then S^{(n)} = S^n / Σ_n is also a MDS.

Experimental results

Research questions

• RQ1 Can the punctual Hilbert scheme S^{[2]} of a K3 surface S have infinite automorphism group when S itself has finite automorphism group?
• RQ2 What geometric conditions on a K3 surface S ensure that Aut(S^{[2]}) is infinite?
• RQ3 Does the Hilbert-Chow morphism μ:S^{[2]}→S^{(2)} preserve the Mori dream space (MDS) property, given that it is an extremal crepant resolution?
• RQ4 Is there a K3 surface S with ρ(S) = 2 and |Aut(S)| < ∞ such that |Aut(S^{[2]})| = ∞?
• RQ5 Under what conditions is the Hilbert scheme S^{[n]} a Mori dream space when the symmetric product S^{(n)} is?

Key findings

• For K3 surfaces S with NS(S) ≅ Λ, the automorphism group Aut(S) is finite, but Aut(S^{[2]}) is infinite, thus affirming Wandel’s question.
• The existence of two non-isomorphic embeddings of S into P^3 induces two distinct Beauville involutions on S^{[2]}, generating an infinite subgroup of Aut(S^{[2]})
• The Hilbert-Chow morphism μ:S^{[2]}→S^{(2)} is an extremal crepant resolution, but S^{[2]} is not a Mori dream space, while S^{(2)} is.
• The nef cone of S^{[2]} is not a finite rational polyhedral cone, which implies that Aut(S^{[2]}) cannot be infinite unless the cone structure is non-polyhedral.
• The symmetric product S^{(2)} is a Mori dream space because S^2 is a MDS and the quotient by Σ_2 preserves the MDS property under the given conditions.
• The construction shows that ρ(S) = 2 is the minimal Picard number for which Aut(S^{[2]}) can be infinite when Aut(S) is finite.