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Julia Schneider

Publications and source records attributed to Julia Schneider.

9 recordsLinked to original sources
arXiv2025

Atomic decompositions for derived categories of G-surfaces

We construct canonical semi-orthogonal decompositions for derived categories of smooth projective surfaces. These decompositions are compatible with the operations in the minimal model program, such as blow-ups and conic bundles. Therefore our construction confirms a conjecture of Kontsevich in dimension two. We work in the G-equivariant setting and over an arbitrary perfect field, and canonical decompositions are consistent with group change and algebraic field extensions. Our method is based on the G-minimal model program for surfaces and on the Sarkisov link factorisation of birational maps between Mori fibre spaces. We characterise rationality of surfaces, and in certain cases, birationality between surfaces in terms of the pieces of these decompositions, which we call atoms.

math.AG↗
arXiv2025

Explicit Sarkisov program for regular surfaces over arbitrary fields and applications

We prove the Sarkisov program for projective surfaces over excellent base rings, including the case of non-perfect base fields $k$ of characteristic $p>0$. We classify the Sarkisov links between Mori fibre spaces and their relations for regular surfaces, generalising work of Iskovskikh. As an application, we discuss rationality problems for regular surfaces and the structure of the plane Cremona group.

math.AG↗
arXiv2024

Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank

We prove that any group of cardinality at most the one of $\mathbb{C}$ is a quotient of any Cremona group of rank at least $4$. This provides a definitive answer to the question of what the quotients of Cremona groups can be. As a consequence, this gives a negative answer to the question of I. Dolgachev of whether Cremona groups of all ranks are generated by involutions. As another application, we show that higher Cremona groups do not enjoy some classical group-theoretic properties (namely, the Hopfian property) which are satisfied by Cremona groups of rank $2$. Finally, we discover that the $3$-torsion of the Cremona group of rank at least $4$ is not countable. To deduce these properties of higher Cremona groups, we first describe the group of birational transformations of a non-trivial Severi-Brauer surface $S$ over a perfect field, proving in particular that if $S$ contains a point of degree $6$, then its group of birational self-maps is not generated by elements of finite order as it admits a surjective group homomorphism to~$\mathbb{Z}$. We then use this result to study Mori fibre spaces over the field of complex numbers, for which the generic fibre is a non-trivial Severi-Brauer surface.

math.AG↗
arXiv2024

Generators of the plane Cremona group over the field with two elements

The plane Cremona group over the finite field $\mathbb{F}_2$ is generated by three infinite families and finitely many birational maps with small base orbits. One family preserves the pencil of lines through a point, the other two preserve the pencil of conics through four points that form either one Galois orbit of size 4, or two Galois orbits of size 2. For each family, we give a generating set that is parametrized by the rational functions over $\mathbb{F}_2$. Moreover, we describe the finitely many remaining maps and give an upper bound on the number needed to generate the Cremona group. Finally, we prove that the plane Cremona group over $\mathbb{F}_2$ is generated by involutions.

math.AG↗
arXiv2023

Algebraic subgroups of the plane Cremona group over a perfect field

We show that any infinite algebraic subgroup of the plane Cremona group over a perfect field is contained in a maximal algebraic subgroup of the plane Cremona group. We classify the maximal groups, and their subgroups of rational points, up to conjugacy by a birational map.

math.AG↗
arXiv2023

CarExpert: Leveraging Large Language Models for In-Car Conversational Question Answering

Large language models (LLMs) have demonstrated remarkable performance by following natural language instructions without fine-tuning them on domain-specific tasks and data. However, leveraging LLMs for domain-specific question answering suffers from severe limitations. The generated answer tends to hallucinate due to the training data collection time (when using off-the-shelf), complex user utterance and wrong retrieval (in retrieval-augmented generation). Furthermore, due to the lack of awareness about the domain and expected output, such LLMs may generate unexpected and unsafe answers that are not tailored to the target domain. In this paper, we propose CarExpert, an in-car retrieval-augmented conversational question-answering system leveraging LLMs for different tasks. Specifically, CarExpert employs LLMs to control the input, provide domain-specific documents to the extractive and generative answering components, and controls the output to ensure safe and domain-specific answers. A comprehensive empirical evaluation exhibits that CarExpert outperforms state-of-the-art LLMs in generating natural, safe and car-specific answers.

cs.CL↗
arXiv2021

Relations in the Cremona group over perfect fields

For perfect fields $k$ satisfying $[\bar k:k]>2$, we construct new normal subgroups of the plane Cremona group and provide an elementary proof of its non-simplicity, following the melody of the recent proof by Blanc, Lamy and Zimmermann that the Cremona group of rank $n$ over (subfields of) the complex numbers is not simple for $n\geq3$.

math.AG↗
arXiv2018

Plane curves of fixed bidegree and their $A_k$-singularities

We provide a tool how one can view a polynomial on the affine plane of bidegree $(a,b)$ - by which we mean that its Newton polygon lies in the triangle spanned by $(a,0)$, $(0,b)$ and the origin - as a curve in a Hirzebruch surface having nice geometric properties. As an application, we study maximal $A_k$-singularities of curves of bidegree $(3,b)$ and find the answer for $b\leq 12$.

math.AG↗