Invariants of Fano varieties in families
We show that the Picard rank is constant in families of Fano varieties (in arbitrary characteristic) and we moreover investigate the constancy of the index.
math.AG↗
arXiv subjects
Ariyan Javanpeykar
Publications and source records attributed to Ariyan Javanpeykar.
The moduli of smooth hypersurfaces with level structure
We construct the moduli space of smooth hypersurfaces with level $N$ structure over $\mathbb{Z}[1/N]$. As an application we show that, for $N$ large enough, the stack of smooth hypersurfaces over $\mathbb{Z}[1/N]$ is uniformisable by a smooth affine scheme. To prove our results, we use the Lefschetz trace formula to show that automorphisms of smooth hypersurfaces act faithfully on their cohomology. We also prove a global Torelli theorem for smooth cubic threefolds over fields of odd characteristic.
math.AG↗
Effectively computing integral points on the moduli of smooth quartic curves
We prove an effective version of the Shafarevich conjecture (as proven by Faltings) for smooth quartic curves. To do so, we establish an effective version of Scholl's finiteness result for smooth del Pezzo surfaces of degree at most four.
math.NT↗
Complete intersections: Moduli, Torelli, and good reduction
We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.
math.AG↗
Belyi's theorem for complete intersections of general type
We give a Belyi-type characterisation of smooth complete intersections of general type over $\mathbb{C}$ which can be defined over $\bar{\mathbb{Q}}$. Our proof uses the higher-dimensional analogue of the Shafarevich boundedness conjecture for families of canonically polarized varieties, finiteness results for maps to varieties of general type, and rigidity theorems for Lefschetz pencils of complete intersections.
math.AG↗
An effective Arakelov-theoretic version of the hyperbolic isogeny theorem
For an integer $e$ and hyperbolic curve $X$ over $\overline{\mathbb Q}$, Mochizuki showed that there are only finitely many isomorphism classes of hyperbolic curves $Y$ of Euler characteristic $e$ with the same universal cover as $X$. We use Arakelov theory to prove an effective version of this finiteness statement.
math.AG↗
Good reduction of algebraic groups and flag varieties
In 1983, Faltings proved that there are only finitely many abelian varieties over a number field of fixed dimension and with good reduction outside a given set of places. In this paper, we consider the analogous problem for other algebraic groups and their homogeneous spaces, such as flag varieties.
math.NT↗
Polynomial bounds for Arakelov invariants of Belyi curves
We explicitly bound the Faltings height of a curve over Q polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings' delta invariant and the self-intersection of the dualizing sheaf. Our results allow us to explicitly bound Arakelov invariants of modular curves, Hurwitz curves and Fermat curves in terms of their genus. Moreover, as an application, we show that the Couveignes-Edixhoven-Bruin algorithm to compute coefficients of modular forms for congruence subgroups of SL2(Z) runs in polynomial time under the Riemann hypothesis for zeta-functions of number fields. This was known before only for certain congruence subgroups. Finally, we use our results to prove a conjecture of Edixhoven, de Jong and Schepers on the Faltings height of a cover of the projective line with fixed branch locus. Our proof uses Merkl's method for bounding Arakelov-Green's functions to deal with archimedean contributions. In the appendix, Peter Bruin proves an explicit version of Merkl's results.
math.AG↗
Szpiro's small points conjecture for cyclic covers
Let $X$ be a smooth, projective and geometrically connected curve of genus at least two, defined over a number field. In 1984, Szpiro conjectured that $X$ has a "small point". In this paper we prove that if $X$ is a cyclic cover of prime degree of the projective line, then $X$ has infinitely many "small points". In particular, we establish the first cases of Szpiro's small points conjecture, including the genus two case and the hyperelliptic case. The proofs use Arakelov theory for arithmetic surfaces and the theory of logarithmic forms.
math.NT↗