## Julia Schneider

I am a postdoc in the group of Stéphane Lamy at the University of Toulouse. I did my PhD in algebraic geometry under the supervision of Jérémy Blanc at the University of Basel.

Interests: Classical algebraic geometry, Cremona group, plane curve singularities, birational geometry, non-closed fields, turtles.

• Generating the plane Cremona group by involutions (with Stéphane Lamy)
We prove that over any perfect field the plane Cremona group is generated by involutions.
• Generators of the plane Cremona group over the field with two elements
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 Jupyter notebook.
 Lemma 4.8 (orbits on projective plane P2) Lemma 4.12 (orbits from quadric Q) Lemma 4.14 and 4.15 (FrobTilde from D5) Lemma 4.18 and Remark 4.19 (orbits from D5) Lemma 4.24 (FrobTilde from D6) Lemma 4.27 (orbits from D6) Corollary (involutions from orbits of size 6 in P2) Corollary (involutions from orbits of size 2 and 3 on D6)
• Algebraic subgroups of the plane Cremona group over a perfect field (with Susanna Zimmermann)
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
Annales de l'Institut Fourier (to appear)
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
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
University of Basel, 2020