Coarse infinite-dimensionality of hyperspaces of finite subsets

We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.

Prelog Chow groups of self-products of degenerations of cubic threefolds

It is unknown whether smooth cubic threefolds have an (integral Chow-theoretic) decomposition of the diagonal, or whether they are stably rational or not in general. As a first step towards making progress on these questions, we compute the (saturated numerical) prelog Chow group of the self-product of a certain degeneration of cubic threefolds.

Lang–Vojta conjecture over function fields for surfaces dominating $${{\mathbb {G}}}_m^2$$

We prove the nonsplit case of the Lang–Vojta conjecture over function fields for surfaces of log general type that are ramified covers of $${{\mathbb {G}}}_m^2$$ G m 2 . This extends the results of Corvaja and Zannier (J Differ Geom 93(3):355–377, 2013), where the conjecture was proved in the split case, and the results of Corvaja and Zannier (J Algebr Geom 17(2):295–333, 2008), Turchet (Trans Amer Math Soc 369(12):8537–8558, 2017) that were obtained in the case of the complement of a degree four and three component divisor in $${{\mathbb {P}}}^2$$ P 2 . We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.

Higher-dimensional Calabi–Yau varieties with dense sets of rational points

We construct higher-dimensional Calabi–Yau varieties defined over a given number field with Zariski dense sets of rational points. We give two elementary constructions in arbitrary dimensions as well as another construction in dimension three which involves certain Calabi–Yau threefolds containing an Enriques surface. The constructions also show that potential density holds for (sufficiently) general members of the families.