Julia Schneider

I am a postdoc at the University of Zurich, in the group of Andrew Kresch.

Before that I was a postdoc at EPFL in the group of Zsolt Patakfalvi, and at the University of Toulouse in the group of Stéphane Lamy. I did my PhD in algebraic geometry under the supervision of Jérémy Blanc at the University of Basel.

Interests: Arithmetic questions on groups of birational transformations, classical algebraic geometry, Cremona groups, plane curve singularities, birational geometry, non-closed fields, turtles.

• Explicit Sarkisov program for regular surfaces over arbitrary fields and applications (with Fabio Bernasconi and Andrea Fanelli and Susanna Zimmermann)
Abstract arXiv
We prove the Sarkisov program for projective surfaces over excellent base rings, including the case of non-perfect base fields 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.
• Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank (with Jérémy Blanc and Egor Yasinsky)
Abstract arXiv
We describe the group of birational transformations of a non-trivial Severi-Brauer surface over a perfect field K, proving in particular that if it contains a point of degree 6, then it is not generated by elements of finite order as it admits a surjective group homomorphism to 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. We prove that any group of cardinality at most the one of C is a quotient of any Cremona group of rank at least 4. As a consequence, this gives a negative answer to the question of Dolgachev of whether the Cremona groups of all ranks are generated by involutions. We also prove that the 3-torsion of the Cremona group of rank at least 4 is not countable.
• Generators of the plane Cremona group over the field with two elements
Abstract arXiv sage code
The plane Cremona group over the finite field 𝔽2 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 𝔽2. 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 𝔽2 is generated by involutions.
The files are sagemath files for JupyterLab: sage-Cremona_over_F2.zip For v4 on arxiv. Work a priori over any finite field.
DPtoolkit.py:some basic functions that will be used throughout
k-structure.py: The information of the minimal del Pezzo surfaces in terms of k-structure.
points_on_P2.ipynb: Compute points in general position on projective plane over any finite field.
points_on_Q.ipynb: as above but for minimal del Pezzo surface of degree 8.
points_on_X5.ipynb: as above but for minimal del Pezzo surface of degree 5.
points_on_X6.ipynb: as above but for minimal del Pezzo surface of degree 6.
Map_P2_66.py: some functions to give explicit equation for the 6:6-link on P2, and the 3:3-link on X6.
Map_P2_55.py: as above but for the 2:2-link on X6.
involution_P2_66.ipynb: Can determine over any field whether the 6:6-link on P2 is an involution, and if yes, find the explicit equation.
involution_X6_22.ipynb: as above but for 2:2-link on X6.
involution_X6_33.ipynb: as above but for 3:3-link on X6.
• Generating the plane Cremona group by involutions (with Stéphane Lamy)
Abstract open access
We prove that over any perfect field the plane Cremona group is generated by involutions.
Algebraic Geometry 11 (1), 111-162, 2024.
• Algebraic subgroups of the plane Cremona group over a perfect field (with Susanna Zimmermann)
Abstract arXiv
EpiGA, vol. 5 (2021), no. 14
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.
• Relations in the Cremona group over perfect fields
Abstract open access
Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 1-42.
For perfect fields k with algebraic closure L satisfying [L: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 ≥ 3.
• Plane curves of fixed bidegree and their A_k-singularities
Abstract open access
North-W. Eur. J. of Math., vol 6, 2020, pages 45-89
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 Ak-singularities of curves of bidegree (3,b) and find the answer for b ≤ 12.
PhD thesis: A birational journey: From plane curve singularities to the Cremona group over perfect fields
open access
University of Basel, 2020