Annales scientifiques de l'École normale supérieure

Moduli of hybrid curves I: Variations of canonical measures

2024-11-18T11:02:16+01:00

We introduce the moduli space of hybrid curves as a new compactification of the moduli space of curves, refining the one obtained by Deligne and Mumford. This is the moduli space for geometric objects which mix complex with higher rank tropical and non-Archimedean geometries, reflecting both discrete and continuous features.

We define canonical measures on hybrid curves which combine and generalize Arakelov– Bergman measures on Riemann surfaces and Zhang measures on metric graphs.

We then show that the universal family of canonically measured hybrid curves over this moduli space varies continuously. This provides a precise link between the non-Archimedean Zhang measure and variations of Arakelov–Bergman measures in families of Riemann sur- faces, answering a question which has been open since the pioneering work of Zhang on admissible pairing in the nineties.

The minimal resolution property for points on generic curves

2024-11-18T11:05:48+01:00

We present an essentially complete solution to the Minimal Resolution Conjecture
for general curves, determining the shape of the minimal resolution of general sets of points on
a general curve C in P^r gree d ≥ 2r. Our methods also provide a proof (valid in arbitrary
characteristic) of the strong version of Butler’s Conjecture on the stability of syzygy bundles on
a general curve of every genus g > 2 in projective space, as well as of the strong (Frobenius)
semistability in positive characteristic of the syzygy bundle of a general curve.

Periodic trivial extension algebras and fractionally Calabi-Yau algebras

2024-11-18T11:06:51+01:00

We study periodicity and twisted periodicity of the trivial extension algebra $T(A)$ of a finite-dimensional algebra $A$. Our main results show that (twisted) periodicity of $T(A)$ is equivalent to $A$ being (twisted) fractionally Calabi--Yau of finite global dimension. We also extend this result to a large class of self-injective orbit algebras.
As a significant consequence, these results give a partial answer to the periodicity conjecture of Erdmann--Skowro\'nski, which expects the classes of periodic and twisted periodic algebras to coincide.
On the practical side, it allows us to construct a large number of new examples of periodic algebras and fractionally Calabi--Yau algebras.
We also establish a connection between periodicity and cluster tilting theory, by showing that twisted periodicity of $T(A)$ is equivalent the $d$-representation-finiteness of the $r$-fold trivial extension algebra $T_r(A)$ for some $r,d\ge 1$. This answers a question by Darp\"o and Iyama.

As applications of our results, we give answers to some other open questions.
We construct periodic symmetric algebras of wild representation type with arbitrary large minimal period, answering a question by Skowro\'nski.
We also show that the class of twisted fractionally Calabi--Yau algebras is closed under derived equivalence, answering a question by Herschend and Iyama.

E_\infty-cells and general linear groups of finite fields

2024-11-18T11:09:19+01:00

We prove new homological stability results for general linear groups over finite fields. These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of $E_\infty$-algebras, guided by computations of homology with coefficients in the $E_1$-split Steinberg module.