# Sets of Special Subvarieties of Bounded Degree

@inproceedings{Urbanik2021SetsOS, title={Sets of Special Subvarieties of Bounded Degree}, author={David Urbanik}, year={2021} }

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…

## 2 Citations

Algebraic Cycle Loci at the Integral Level

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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…

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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…

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