New perspectives on categorical Torelli theorems for del Pezzo threefolds

Let $Y_d$ be a del Pezzo threefold of Picard rank one and degree $d\geq 2$. In this paper, we apply two different viewpoints to study $Y_d$ via a particular admissible subcategory of its bounded derived category, called the Kuznetsov component: (i) Brill-Noether reconstruction. We show that $Y_d$ can be uniquely recovered as a Brill-Noether locus of Bridgeland stable objects in its Kuznetsov component. (ii) Exact equivalences. We prove that, up to composing with an explicit auto-equivalence, any Fourier-Mukai type equivalence of Kuznetsov components of two del Pezzo threefolds of degree $2\leq d\leq 4$ can be lifted to an equivalence of their bounded derived categories. As a result, we obtain a complete description of the group of Fourier-Mukai type auto-equivalences of the Kuznetsov component of $Y_d$. We also describe the group of Fourier-Mukai type auto-equivalences of Kuznetsov components of index one prime Fano threefolds $X_{2g-2}$ of genus $g=6$ and $8$. As an application, first we identify the group of automorphisms of $X_{14}$ and its associated $Y_3$. Then we give a new disproof of Kuznetsov's Fano threefold conjecture by assuming Gushel-Mukai threefolds are general. In an appendix, we classify instanton sheaves on quartic double solids, generalizing a result of Druel.Comment: 35 pages, added results on index one prime Fano threefolds and the application to Kuznetsov's Fano threefold conjecture. Comments are very welcome

Getting More out of Large Language Models for Proofs

Large language models have the potential to simplify formal theorem proving and make it more accessible. But how to get the most out of these models is still an open question. To answer this question, we take a step back and explore the failure cases of these models using common prompting-based techniques. Our talk will discuss these failure cases and what they can teach us about how to get more out of these models

SwinFIR: Revisiting the SwinIR with Fast Fourier Convolution and Improved Training for Image Super-Resolution

Transformer-based methods have achieved impressive image restoration performance due to their capacities to model long-range dependency compared to CNN-based methods. However, advances like SwinIR adopts the window-based and local attention strategy to balance the performance and computational overhead, which restricts employing large receptive fields to capture global information and establish long dependencies in the early layers. To further improve the efficiency of capturing global information, in this work, we propose SwinFIR to extend SwinIR by replacing Fast Fourier Convolution (FFC) components, which have the image-wide receptive field. We also revisit other advanced techniques, i.e, data augmentation, pre-training, and feature ensemble to improve the effect of image reconstruction. And our feature ensemble method enables the performance of the model to be considerably enhanced without increasing the training and testing time. We applied our algorithm on multiple popular large-scale benchmarks and achieved state-of-the-art performance comparing to the existing methods. For example, our SwinFIR achieves the PSNR of 32.83 dB on Manga109 dataset, which is 0.8 dB higher than the state-of-the-art SwinIR method

Brill--Noether theory for Kuznetsov components and refined categorical Torelli theorems for index one Fano threefolds

We show by a uniform argument that every index one prime Fano threefold $X$ of genus $g\geq 6$ can be reconstructed as a Brill--Noether locus inside a Bridgeland moduli space of stable objects in the Kuznetsov component $\mathcal{K}u(X)$. As an application, we prove refined categorical Torelli theorems for $X$ and compute the fiber of the period map for each Fano threefold of genus $g\geq 7$ in terms of a certain gluing object associated with the subcategory $\langle \mathcal{O}_X \rangle^{\perp}$. This unifies results of Mukai, Brambilla-Faenzi, Debarre-Iliev-Manivel, Faenzi-Verra, Iliev-Markushevich-Tikhomirov and Kuznetsov.