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Volume 10
Dixième volume de l'Épijournal de Géométrie Algébrique — 2026
1. Real plane separating (M-2)-curves of degree d and totally real pencils of degree d-3
It is well known that a non-singular real plane projective curve of degree five with five connected components is separating if and only if its ovals are in non-convex position. In this article, this property is set into a different context and generalised to all real plane separating (M-2)-curves.
2. $G$-torsors on perfectoid spaces
For any rigid analytic group variety $G$ over a non-archimedean field $K$ over $\mathbb Q_p$, we study $G$-torsors on adic spaces over $K$ in the $v$-topology. Our main result is that on perfectoid spaces, $G$-torsors in the étale and $v$-topology are equivalent. This generalises the known cases of $G=\mathbb G_a$ and $G=\mathrm{GL}_n$ due to Scholze and Kedlaya--Liu. On a general adic space $X$ over $K$, where there can be more $v$-topological $G$-torsors than étale ones, we show that for any open subgroup $U\subseteq G$, any $G$-torsor on $X_v$ admits a reduction of structure group to $U$ étale-locally on $X$. This has applications in the context of the $p$-adic Simpson correspondence: For example, we use it to show that on any adic space, generalised $\mathbb Q_p$-representations are equivalent to $v$-vector bundles.
3. Gromov-Witten and Welschinger invariants of del Pezzo varieties
In this paper, we establish formulas for computing genus-$0$ Gromov-Witten and Welschinger invariants of some del Pezzo varieties of dimension three by comparing to that of dimension two. These formulas are generalizations of that given in three-dimensional projective space by E. Brugallé and P. Georgieva in 2016.
4. Algebraic hyperbolicity of very general hypersurfaces in homogeneous varieties
We generalize techniques by Coskun, Riedl, and Yeong, and obtain an almost optimal bound on the degree for the algebraic hyperbolicity of very general hypersurfaces in rational homogeneous varieties. As examples, we work out the cases of very general hypersurfaces in Grassmannians and products therefore, orthogonal and symplectic Grassmannians, and flag varieties.