arXiv subjects

Ippei Nagamachi

Publications and source records attributed to Ippei Nagamachi.

7 recordsLinked to original sources

arXiv2022

Topics in the Grothendieck conjecture for hyperbolic polycurves of dimension 2

In this paper, we study the anabelian geometry of hyperbolic polycurves of dimension 2 over sub-p-adic fields. In 1-dimensional case, Mochizuki proved the Hom version of the Grothendieck conjecture for hyperbolic curves over sub-p-adic fields and the pro-p version of this conjecture. In 2-dimensional case, a naive analogue of this conjecture does not hold for hyperbolic polycurves over general sub-p-adic fields. Moreover, the Isom version of the pro-p Grothendieck conjecture does not hold in general. We explain these two phenomena and prove the Hom version of the Grothendieck conjecture for hyperbolic polycurves of dimension 2 under the assumption that the Grothendieck section conjecture holds for some hyperbolic curves.

math.NT↗

arXiv2022

Minimal log regular models of hyperbolic curves over discrete valuation fields

In the famous paper of Deligne and Mumford, they proved that a proper hyperbolic curve over a discrete valuation field has stable reduction if and only if the Jacobian variety of the curve has stable reduction in the case where the residue field of its valuation ring is algebraically closed. In the proof, the theory of minimal regular models played an important role. In this paper, we establish a theory of minimal log regular models of curves. As a key tool for this theory, we give a criterion for $2$-dimensional local schemes to be log regular in terms of their minimal desingularization. Moreover, as an application of this theory, we prove the above equivalence without the assumption on the residue field.

math.NT↗

arXiv2022

On behavior of conductors, Picard schemes, and Jacobian numbers of varieties over imperfect fields

Let $X$ be a regular geometrically integral variety over an imperfect field $K$. Unlike the case of characteristic $0$, $X':=X\times_{\mathrm{Spec}\,K}\mathrm{Spec}\,K'$ may have singular points for a (necessarily inseparable) field extension $K'/K$. In this paper, we define new invariants of the local rings of codimension $1$ points of $X'$, and use these invariants for the calculation of $δ$-invariants (, which relate to genus changes,) and conductors of such points. As a corollary, we give refinements of Tate's genus change theorem and the Patakfalvi-Waldron Theorem. Moreover, when $X$ is a curve, we show that the Jacobian number of $X$ is $2p/(p-1)$ times of the genus change by using the above calculation. In this case, we also relate the structure of the Picard scheme of $X$ with invariants of singular points of $X$. To prove such a relation, we give a characterization of the geometrical normality of algebras over fields of positive characteristic.

math.AG↗

arXiv2021

On homotopy exact sequences for normal schemes

Consider a morphism between connected locally Noetherian normal schemes. In this paper, we discuss when the sequence of the etale fundamental groups associated to the morphism is exact. Moreover, we give a characterization of when the kernel of the induced homomorphism between their fundamental groups is topologically finitely generated, for the morphism from a smooth variety to a smooth curve.

math.NT↗

arXiv2020

Criteria for good reduction of proper polycurves

We give good reduction criteria for hyperbolic polycurves, i.e., successive extensions of families of curves, under mild assumption. These criteria are higher dimensional version of the good reduction criterion for hyperbolic curves given by Oda and Tamagawa.

math.NT↗

arXiv2019

The Shafarevich conjecture and some extension theorems for proper hyperbolic polycurves

In this paper, we prove the Shafarevich conjecture for proper hyperbolic polycurves, which is a higher dimensional analogue of that for proper hyperbolic curves. First, we study theories of proper hyperbolic polycurves over regular schemes. For example, we generalize the moduli theory of Kodaira fibrations due to Jost and Yau. We also show the Neron property of proper smooth models of proper hyperbolic polycurves over Dedekind schemes under an assumption on residual characteristics. We then apply these extension theories to the proof of the Shafarevich conjecture for proper hyperbolic polycurves.

math.NT↗

arXiv2016

On a good reduction criterion for proper polycurves with sections

We give a good reduction criterion for proper polycurves with sections,i.e., successive extensions of family of curves with section, under mild assumption. This criterion is a higher dimensional version of the good reduction criterion for hyperbolic curves given by Oda and Tamagawa.

math.NT↗