10th Conference on Arithmetic Algebraic Geometry

This talk is based on a joint paper with Momonari Kudo (arXiv 1607.01114), where we proved that there is no superspecial curve of genus 4 in characteristic 7. This is an answer to the genus 4 case of the problem proposed by Ekedahl in 1987. This implies the non-existence of maximal curve of genus 4 over F49, which updated the table at manypoints.org. We give an algorithm to enumerate superspecial nonhyperelliptic curves of genus 4 in arbitrary p ≥ 5, and for p ≤ 7 we excute it with our implementation on a computer algebra system Magma. Our result in p = 5 re-proves the uniqueness of maximal curves of genus 4 over F25. Takehiko Yasuda (Osaka. Univ.) Vojta’s conjecture and singularities Abstract. Vojta’s conjecture was originally formulated for the pair of a nonsingular variety and a normal crossing divisor. In a recent paper, he generalized it further by replacing the normal crossing divisor by an arbitrary closed subscheme. In this talk, I will present a further generalization to log pairs with a possibly singular ambient variety. In this generalization, we use variants of multiplier ideals. In the case of a singular variety with a big canonical divisor, the generalized conjecture roughly says that most rational points are located near the non-canonical locus. Vojta’s conjecture was originally formulated for the pair of a nonsingular variety and a normal crossing divisor. In a recent paper, he generalized it further by replacing the normal crossing divisor by an arbitrary closed subscheme. In this talk, I will present a further generalization to log pairs with a possibly singular ambient variety. In this generalization, we use variants of multiplier ideals. In the case of a singular variety with a big canonical divisor, the generalized conjecture roughly says that most rational points are located near the non-canonical locus. Sho Tanimoto (Univ. Copenhagen) On the geometry of exceptional sets in Manin’s conjecture Abstract. Manin’s conjecture predicts an asymptotic formula for the counting function of rational points on a Fano variety after removing the contribution from an exceptional set. In this talk, I would like to discuss the geometry of this exceptional set using birational geometry, e.g., the minimal model program. This is joint work with Brian Lehmann. Manin’s conjecture predicts an asymptotic formula for the counting function of rational points on a Fano variety after removing the contribution from an exceptional set. In this talk, I would like to discuss the geometry of this exceptional set using birational geometry, e.g., the minimal model program. This is joint work with Brian Lehmann. Antoine Ducros (Pris 6) Skeletons of Berkovich spaces Abstract. One of the most interesting features of Berkovich theory is the existence, inside Berkovich analytic spaces, of subsets that inherit from the ambient analytic structure a piecewise-linear structure: the so-called /skeletons/. In this talk, I plan to make a general survey on skeletons, including recent results (joint work in progress with Amaury Thuillier) about their behavior under inverse and direct images. One of the most interesting features of Berkovich theory is the existence, inside Berkovich analytic spaces, of subsets that inherit from the ambient analytic structure a piecewise-linear structure: the so-called /skeletons/. In this talk, I plan to make a general survey on skeletons, including recent results (joint work in progress with Amaury Thuillier) about their behavior under inverse and direct images.