arXiv subjects
Publications and source records attributed to Michał Szachniewicz.
We use generalised Bialynicki-Birula decomposition, apolarity and obstruction theories to prove non-reducedness of the Hilbert scheme of $13$ points on $\mathbb{A}^6$. Our argument doesn't involve computer calculations and gives an example of a fractal-like structure on this Hilbert scheme.
We present foundations for globally valued fields, a class of enriched fields capturing some aspects of the geometry of global fields, based on the product formula. We show that this class is axiomatizable in the framework of unbounded continuous logic, opening the way to decidability and transfer theorems. We provide a dictionary between various data defining such extra structure: syntactic, Arakelov or cycle-theoretic, and measure theoretic. In particular we obtain a representation theorem relating globally valued fields and adelic curves defined by Chen and Moriwaki.
We study the complexity of the $KK$-equivalence relation on unital $C^*$-algebras, in the sense of descriptive set theory. We prove that $KK$-equivalence is analytic, which in turn shows that the set of separable $C^*$-algebras satisfying the UCT is analytic. This allows us to prove a game-theoretic refinement of the unital classification theorem: there is a transfer of strategies between Ehrenfeucht-Fraïssé games (of various lengths) on classifiable $C^*$-algebras and their invariants.
We study model-complete fields that avoid a given quasi-project variety $V$. There is a close connection between hyperbolicity of $V$ and the existence of the model companion for the theory of characteristic-zero fields avoiding rational points on $V$. This gives a model theoretic notion of hyperbolicity that we call excludability. In particular, we show that if $V$ is a Brody hyperbolic projective variety over $\mathbb{Q}$ with $V(\mathbb{Q}) = \varnothing$, then the model companion, called $V\XF$, exists. We also study some model-theoretic properties of $V\mathrm{XF}$. This extends the results for curves by Will Johnson and the second author.
We study the variation of heights of cycles in flat families over number fields or, more generally, globally valued fields. To a finite type scheme over a GVF we associate a locally compact Hausdorff space which we refer to as its GVF analytification. For a flat projective family, we prove that the height of fibres is a continuous function on the GVF analytification of the base. As an application, we prove Roberto Gualdi's conjecture on limit heights of complete intersections in toric varieties.
Enhancing low-light images while maintaining natural colors is a challenging problem due to camera processing variations and limited access to photos with ground-truth lighting conditions. The latter is a crucial factor for supervised methods that achieve good results on paired datasets but do not handle out-of-domain data well. On the other hand, unsupervised methods, while able to generalize, often yield lower-quality enhancements. To fill this gap, we propose Dimma, a semi-supervised approach that aligns with any camera by utilizing a small set of image pairs to replicate scenes captured under extreme lighting conditions taken by that specific camera. We achieve that by introducing a convolutional mixture density network that generates distorted colors of the scene based on the illumination differences. Additionally, our approach enables accurate grading of the dimming factor, which provides a wide range of control and flexibility in adjusting the brightness levels during the low-light image enhancement process. To further improve the quality of our results, we introduce an architecture based on a conditional UNet. The lightness value provided by the user serves as the conditional input to generate images with the desired lightness. Our approach using only few image pairs achieves competitive results compared to fully supervised methods. Moreover, when trained on the full dataset, our model surpasses state-of-the-art methods in some metrics and closely approaches them in others.
Self-supervised methods have been proven effective for learning deep representations of 3D point cloud data. Although recent methods in this domain often rely on random masking of inputs, the results of this approach can be improved. We introduce PointCAM, a novel adversarial method for learning a masking function for point clouds. Our model utilizes a self-distillation framework with an online tokenizer for 3D point clouds. Compared to previous techniques that optimize patch-level and object-level objectives, we postulate applying an auxiliary network that learns how to select masks instead of choosing them randomly. Our results show that the learned masking function achieves state-of-the-art or competitive performance on various downstream tasks. The source code is available at https://github.com/szacho/pointcam.
We use the differentiability of the arithmetic volume function and an arithmetic Bertini type theorem to classify when one can find a closed point on the generic fiber of an arithmetic variety, whose heights with respect to some finite tuple of arithmetic $\mathbb{R}$-divisors approximate a given tuple of real numbers. We use this result to prove existential closedness of $\overline{\mathbb{Q}}$ as a globally valued field (abbreviated GVF). We introduce GVF functionals on the space of arithmetic $\mathbb{R}$-divisors and interpret the essential infimum function as the infimum of values of normalised GVF functionals, at least when the generic part of the arithmetic $\mathbb{R}$-divisor is big. We also give a new criterion on equality in one of the Zhang's inequalities.