arXiv subjects
Publications and source records attributed to Amaël Broustet.
Let $f\colon X\to X$ be a polarised endomorphism of a complex projective manifold, and let $D\subset X$ be a reduced totally invariant hypersurface. We prove that every irreducible component of the singular locus of $D$ has codimension one in $D$. In particular, if $D$ is normal, then it is smooth. As an application we prove the linearity of totally invariant divisors for endomorphisms of $\mathbb P^4$.
Persistent homology (PH) has recently emerged as a powerful tool for extracting topological features. Integrating PH into machine learning and deep learning models enhances topology awareness and interpretability. However, most PH methods on graphs rely on a limited set of filtrations, such as degree-based or weight-based filtrations, which overlook richer features like recurring information across the dataset and thus restrict expressive power. In this work, we propose a novel graph filtration called Frequent Subgraph Filtration (FSF), which is derived from frequent subgraphs and produces stable and information-rich frequency-based persistent homology (FPH) features. We study the theoretical properties of FSF and provide both proofs and experimental validation. Beyond persistent homology itself, we introduce two approaches for graph classification: an FPH-based machine learning model (FPH-ML) and a hybrid framework that integrates FPH with graph neural networks (FPH-GNNs) to enhance topology-aware graph representation learning. Our frameworks bridge frequent subgraph mining and topological data analysis, offering a new perspective on topology-aware feature extraction. Experimental results show that FPH-ML achieves competitive or superior accuracy compared with kernel-based and degree-based filtration methods. When integrated into graph neural networks, FPH yields relative performance gains ranging from 0.4 to 21 percent, with improvements of up to 8.2 percentage points over GCN and GIN backbones across benchmarks.
Let X be a smooth projective threefold, and let A be an ample line bundle such that $K_X+A$ is nef. We show that if $K_X$ or $-K_X$ is pseudoeffective, the adjoint bundle $K_X+A$ has global sections. We also give a very short proof of the Beltrametti-Sommese conjecture in dimension three, recently proven by Fukuma: if A is an ample line bundle such that $K_X+2A$ is nef, the adjoint bundle $K_X+2A$ has global sections.
Let X be a normal variety such that $K_X$ is Q-Cartier, and let $f: X \rightarrow X$ be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that are not log-canonical and the dynamics of the endomorphism f. As a consequence we prove that if X is projective and f polarised, then X has at most log-canonical singularities.
We discuss the Calabi--Yau type structure of normal projective surfaces and Mori dream spaces admitting a non-trivial polarized endomorphism.
It is well known that a smooth projective Fano variety is rationally connected. Recently Zhang (and later Hacon and McKernan as a special case of their work on the Shokurov RC-conjecture) proved that the same conclusion holds for a klt pair $(X,\D)$ such that $-(K_X+\D)$ is big and nef. We prove here a natural generalization of the above result by dropping the nefness assumption. Namely we show that a klt pair $(X,\D)$ such that $-(K_X+\D)$ is big is rationally connected modulo the non-nef locus of $-(K_X+\D)$. This result is a consequence of a more general structure theorem for arbitrary pairs $(X,\D)$ with $-(K_X+\D)$ pseff.
We prove that Seshadri constants of some ample divisors are bigger than 1 on smooth threefolds whose anticanonical bundle is nef or on Fano varieties of small coindice. The main tools are (some known cases of) the Kawamata's effective non-vanishing conjecture and the adjunction theory. We prove the non-vanishing conjecture in dimension 3 in the case of line bundles of "high" volume using Kawamata's subadjunction formula.
We compute in this note the Seshadri constants of the anticanonical bundle at every point of Del Pezzo surfaces. During the proof, we enlight the role of rational curves in our computations. We present then two exemples where the positivity of the anticanonical bundle cannot be detected using rational curves.