• Corpus ID: 237532441

Sets of Special Subvarieties of Bounded Degree

  title={Sets of Special Subvarieties of Bounded Degree},
  author={David Urbanik},
Let f : X Ñ S be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base S, and let V “ R2kf ̊Zpkq be the integral variation of Hodge structure coming from degree 2k cohomology it induces. Associated to V one has the so-called Hodge locus HLpSq Ă S, which is a countable union of “special” algebraic subvarieties of S parametrizing those fibres of V possessing extra Hodge tensors (and so conjecturally, those fibres of f possessing extra algebraic cycles… 
Algebraic Cycle Loci at the Integral Level
Let f : X Ñ S be a smooth projective family defined over O K r S ´ 1 s , where K Ă C is a number field and S is a finite set of primes. For each prime p P O K r S ´ 1 s with residue field κ p p q , we
Effective Methods for Diophantine Finiteness
Let K Ă C be a number field, and let OK,N “ OKrN s be its ring of N-integers. Recently, Lawrence and Venkatesh proposed a general strategy for proving the Shafarevich conjecture for the fibres of a


On the locus of Hodge classes
Let S be a complex algebraic variety and {Xs}s∈S a family of non singular projective varieties parametrized by S: the Xs are the fibers of f : X → S, with X projective and smooth over S. Fix s ∈ S,
On the fields of definition of Hodge loci.
A polarizable variation of Hodge structure over a smooth complex quasi projective variety $S$ is said to be defined over a number field $L$ if $S$ and the algebraic connection associated to the
Fields of definition of Hodge loci
We show that an irreducible component of the Hodge locus of a polarizable variation of Hodge structure of weight 0 on a smooth complex variety X is defined over an algebraically closed subfield k of
On computing absolutely irreducible components of algebraic varieties with parameters
  • A. Ayad
  • Mathematics, Computer Science
  • 2010
This paper presents a new algorithm for computing absolutely irreducible components of n-dimensional algebraic varieties defined implicitly by parametric homogeneous polynomial equations over
Effective computations for weakly optimal subvarieties
Ren and the second author established that the weakly optimal subvarieties (e.g. maximal weakly special subvarieties) of a subvariety V of a Shimura variety arise in finitely many families. In this
This note was produced as part of the Stacks Project workshop in August 2017, under the guidance of Brian Conrad. Briefly, the process of Weil restriction is the algebro-geometric analogue of viewing
An algorithmic approach to Chevalley's Theorem on images of rational morphisms between affine varieties
A new constructive geometric proof of the affine version of Chevalley's Theorem is introduced and an efficient code for computing the constructible image of rational maps between affine varieties is implemented.
Applications of the hyperbolic Ax–Schanuel conjecture
In 2014, Pila and Tsimerman gave a proof of the Ax–Schanuel conjecture for the $j$ -function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We
Commutative Algebra: Constructive Methods: Finite Projective Modules
Translated from the popular French edition, this book offers a detailed introduction to various basic concepts, methods, principles, and results of commutative algebra. It takes a constructive