Measuring birational derived splinters

1. Problem Statement and Motivation

The paper addresses the challenge of quantifying and studying singularities in algebraic geometry, specifically extending the concept of rational singularities (well-understood in characteristic zero) to arbitrary characteristics, including positive and mixed characteristics.

• Context: The authors build upon the theory of splinters and derived splinters.
  • A scheme $X$ is a splinter if the natural map $\mathcal{O}_X \to f_*\mathcal{O}_Y$ splits for all finite surjective morphisms $f: Y \to X$.
  • A scheme $X$ is a derived splinter if the map $\mathcal{O}_X \to Rf_*\mathcal{O}_Y$ splits for all proper surjective morphisms.
  • A scheme $X$ is a birational derived splinter if $\mathcal{O}_X \to Rf_*\mathcal{O}_Y$ splits for all proper birational morphisms $f: Y \to X$.
• The Gap: While derived splinters coincide with rational singularities in characteristic zero, the behavior in positive characteristic is complex (e.g., elliptic curves in positive characteristic are birational derived splinters but not derived splinters). Existing methods often rely on vanishing theorems (like Grauert–Riemenschneider) that fail in positive characteristic.
• Goal: The authors aim to introduce a numerical invariant to measure the "failure" of a scheme to be a birational derived splinter and to study how this invariant behaves under geometric operations (base change, products, localization) without relying on characteristic-specific vanishing theorems.

2. Methodology: Categorical Generation and "Level"

The core methodology relies on categorical techniques within the framework of triangulated categories, specifically the derived category of coherent sheaves, $D^b_{\text{coh}}(X)$.

• Generation of Triangulated Categories: The authors utilize the notion of generation introduced by Balmer, Van den Bergh, and others. For a subcategory $\mathcal{S} \subseteq \mathcal{T}$, they define $\langle \mathcal{S} \rangle_n$ as the subcategory generated by $\mathcal{S}$ using shifts, direct sums, direct summands, and at most $n$ cones.
• The Invariant "Level": For an object $E \in \mathcal{T}$, the level of $E$ with respect to $\mathcal{S}$, denoted $\text{level}_{\mathcal{S}}(E)$, is the smallest integer $N$ such that $E \in \langle \mathcal{S} \rangle_N$. If no such $N$ exists, the level is $\infty$.
• The New Invariant ($\mu_{\text{bds}}$):
  The authors define a numerical invariant $\mu_{\text{bds}}(X)$ for a Noetherian scheme $X$:
  $$\mu_{\text{bds}}(X) := \sup \{ \text{level}_{Rf_* D^b_{\text{coh}}(Y)}(\mathcal{O}_X) \mid f: Y \to X \text{ is proper and birational} \}$$
  • Interpretation: $\mu_{\text{bds}}(X) = 1$ if and only if $X$ is a birational derived splinter. Higher values indicate a greater "failure" to satisfy the splitting condition.
  • Key Lemma (1.2): The splitting of $\mathcal{O}_X \to Rf_*\mathcal{O}_Y$ is equivalent to $\mathcal{O}_X$ having level 1 with respect to the image of the derived category of $Y$.

3. Key Contributions and Results

The paper establishes several structural properties of $\mu_{\text{bds}}$ and its relationship to geometric operations.

A. Characterization via Blowups and Resolutions

• Reduction to Blowups: The invariant can be computed by considering only blowups along nonzero ideal sheaves (Proposition 3.3).
• Existence of Resolution: If $X$ admits a resolution of singularities $f: \tilde{X} \to X$, then $\mu_{\text{bds}}(X)$ is finite and equals the level of $\mathcal{O}_X$ with respect to $Rf_* D^b_{\text{coh}}(\tilde{X})$ (Theorem 3.4). This independence from the specific resolution is crucial.

B. Local Behavior

• Stalk-wise Calculation: For an integral normal affine scheme $X = \text{Spec}(R)$ admitting a resolution, the global invariant is the supremum of the local invariants at all points:
  $$\mu_{\text{bds}}(X) = \sup_{p \in X} \{ \mu_{\text{bds}}(\mathcal{O}_{X,p}) \}$$
  (Proposition 3.12). This clarifies the local nature of the singularity measurement in the presence of resolutions.
• Completion and Localization: The invariant behaves predictably under completion ($\mu_{\text{bds}}(R) \geq \mu_{\text{bds}}(\hat{R})$) and localization ($\mu_{\text{bds}}(S^{-1}R) \leq \mu_{\text{bds}}(R)$).

C. Behavior under Geometric Operations (Over Perfect Fields)

The authors prove how $\mu_{\text{bds}}$ behaves under base change and products, assuming the existence of resolutions of singularities and working over a perfect field $k$.

1. Base Change:

   • $\mu_{\text{bds}}(X) \geq \mu_{\text{bds}}(X \times_k \text{Spec}(L))$ for any field extension $L/k$.
   • If $L/k$ is finite, equality holds.
   • Corollary: $X$ is a birational derived splinter if and only if $X \times_k \bar{k}$ is one (Theorem 4.4). This establishes that the property is preserved/reflected by algebraic closure.

2. Products:

   • For proper schemes $X, Y$ over $k$:
     $$\max\{\mu_{\text{bds}}(X), \mu_{\text{bds}}(Y)\} \leq \mu_{\text{bds}}(X \times_k Y) \leq \mu_{\text{bds}}(X) \cdot \mu_{\text{bds}}(Y)$$
   • Corollary: $X$ and $Y$ are birational derived splinters if and only if their product $X \times_k Y$ is one (Corollary 4.7).

D. Connection to Pseudo-rationality

The paper connects birational derived splinters to pseudo-rational singularities. For Cohen-Macaulay schemes with a dualizing complex, being a birational derived splinter is equivalent to being pseudo-rational (Lemma 2.10). This bridges the gap between derived categorical definitions and classical singularity theory.

4. Significance and Impact

• Characteristic Independence: Unlike classical approaches to rational singularities which often fail in positive characteristic due to the lack of vanishing theorems, this categorical approach works uniformly across characteristics (zero, positive, and mixed).
• Quantification of Singularities: The introduction of $\mu_{\text{bds}}$ provides a concrete numerical measure for the "severity" of a singularity relative to the birational derived splinter property, moving beyond a binary "yes/no" classification.
• New Tools for Positive Characteristic: The results regarding products and base change in positive characteristic are novel. They provide a robust framework for studying singularities in settings where traditional tools (like Kovács' splitting criteria) are unavailable.
• Categorical Perspective: The paper reinforces the philosophy that geometric properties of varieties are deeply reflected in the structure of their derived categories. It demonstrates how "generation" in triangulated categories can be used as a precise tool for measuring geometric failure.

In summary, this work successfully generalizes the study of rational singularities to a broader context using derived categorical methods, providing a new invariant ($\mu_{\text{bds}}$) that quantifies singularities and behaves well under standard geometric operations, even in the challenging landscape of positive characteristic.