[Paper Review] The geometry of degenerations of Hilbert schemes of points

Martin G. Gulbrandsen, Lars Halvard Halle+2 more|arXiv (Cornell University)|Feb 2, 2018

Advanced Algebra and Geometry21 references3 citations

TL;DR

This paper constructs and studies a degeneration $ I^n_{X/C} \to C $ of the relative Hilbert scheme of $ n $ points on a strict simple degeneration $ f: X \to C $, showing that when $ \dim X_c \leq 2 $, the degeneration is a dlt model and, if $ f $ is good, a good minimal dlt model. It computes the dual complex of the central fiber and relates it to the essential skeleton of the generic fiber, and proves that for $ K3 $ surface degenerations, the stack $ \mathcal{I}^n_{X/C} $ carries a nowhere degenerate relative logarithmic 2-form.

ABSTRACT

Given a strict simple degeneration $f \colon X o C$ the first three authors previously constructed a degeneration $I^n_{X/C} o C$ of the relative degree $n$ Hilbert scheme of $0$-dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of $f$ is at most $2$. In this case we show that $I^n_{X/C} o C$ is a dlt model. This is even a good minimal dlt model if $f \colon X o C$ has this property. We compute the dual complex of the central fibre $(I^n_{X/C})_0$ and relate this to the essential skeleton of the generic fibre. For a type II degeneration of $K3$ surfaces we show that the stack ${\mathcal I}^n_{X/C} o C$ carries a nowhere degenerate relative logarithmic $2$-form. Finally we discuss the relationship of our degeneration with the constructions of Nagai.

Motivation & Objective

• To understand the birational and singular geometry of the degeneration $ I^n_{X/C} \to C $ of the relative Hilbert scheme of $ n $ points on a strict simple degeneration $ f: X \to C $.
• To determine whether $ I^n_{X/C} \to C $ is a dlt model when the fiber dimension of $ f $ is at most 2.
• To compute the dual complex of the central fiber $ (I^n_{X/C})_0 $ and relate it to the essential skeleton of the generic fiber.
• To investigate the existence of a nowhere degenerate relative logarithmic 2-form on $ \mathcal{I}^n_{X/C} \to C $ for type II degenerations of $ K3 $ surfaces.
• To compare the constructed degeneration $ I^n_{X/C} $ with Nagai's construction via symmetric products and toric methods.

Proposed method

• Constructs $ I^n_{X/C} $ as a GIT quotient of the relative Hilbert scheme $ \mathrm{Hilb}^n(X[n]/C[n]) $ by the torus $ G[n] \cong (\mathbb{G}_m)^n $, using Li's expanded degeneration $ X[n] \to C[n] $.
• Uses the fact that GIT semi-stable subschemes are supported only on the smooth locus of fibers, enabling control via combinatorial stability conditions.
• Proves that the GIT quotient $ I^n_{X/C} $ is normal and $ \mathbb{Q} $-factorial by analyzing the $ G[n] $-action and stabilizer groups.
• Establishes the dlt property of $ I^n_{X/C} \to C $ by showing that the pair $ (I^n_{X/C}, (I^n_{X/C})_0) $ satisfies the dlt condition via resolution and discrepancy analysis.
• Computes the dual complex of $ (I^n_{X/C})_0 $ by analyzing the combinatorics of the central fiber's irreducible components and their intersections.
• Uses the $ \mathbb{Z}_2 $-equivariance of the blow-up resolution and $ G[2] $-invariant functions to show that $ \mathrm{Bl}_T(X \times_C X) \to X \times_C X $ is isomorphic to $ I^2_{X/C} $, proving the isomorphism in the $ n=2 $ case.

Experimental results

Research questions

• RQ1 Is the degeneration $ I^n_{X/C} \to C $ a dlt model when the fiber dimension of $ f: X \to C $ is at most 2?
• RQ2 When is $ I^n_{X/C} \to C $ a good minimal dlt model, i.e., when is $ K_{I^n_{X/C}} + (I^n_{X/C})_0 $ semi-ample over $ C $?
• RQ3 What is the structure of the dual complex of the central fiber $ (I^n_{X/C})_0 $, and how does it relate to the essential skeleton of the generic fiber?
• RQ4 Does the stack $ \mathcal{I}^n_{X/C} \to C $ carry a nowhere degenerate relative logarithmic 2-form in the case of type II degenerations of $ K3 $ surfaces?
• RQ5 How does the GIT-based construction of $ I^n_{X/C} $ compare with Nagai's toric construction via symmetric products?

Key findings

• For $ \dim X_c \leq 2 $, the degeneration $ I^n_{X/C} \to C $ is a dlt model: $ I^n_{X/C} $ is normal and $ \mathbb{Q} $-factorial, and the special fiber $ (I^n_{X/C})_0 $ is a reduced divisor.
• If $ f: X \to C $ is a good semi-stable degeneration, then $ I^n_{X/C} \to C $ is a good minimal dlt model, i.e., $ K_{I^n_{X/C}} + (I^n_{X/C})_0 $ is semi-ample over $ C $.
• The dual complex of the central fiber $ (I^n_{X/C})_0 $ is computed and shown to be isomorphic to the dual complex of the central fiber of the expanded degeneration $ X[n] \to C[n] $.
• For type II degenerations of $ K3 $ surfaces, the stack $ \mathcal{I}^n_{X/C} \to C $ carries a nowhere degenerate relative logarithmic 2-form, which is a key geometric invariant.
• In the case $ n=2 $, the paper proves that $ I^2_{X/C} \cong \mathrm{Bl}_T(X \times_C X) $, where $ T $ is the singular locus $ X_0^{\text{sing}} \times X_0^{\text{sing}} $, via an explicit $ G[2] $-equivariant isomorphism.
• The construction of $ I^n_{X/C} $ via GIT is shown to be isomorphic to Nagai's construction via symmetric products and toric geometry, confirming consistency across different approaches.