Let k be a field, A be a finitely generated associative k-algebra and Rep A [n] be the functor Fields k → Sets, which sends a field K containing k to the set of isomorphism classes of representations of A K of dimension at most n. We study the asymptotic behavior of the essential dimension of this functor, i.e., the function r A (n) := ed k (Rep A [n]), as n → ∞. In particular, we show that the rate of growth of r A (n) determines the representation type of A. That is, r A (n) is bounded from above if A is of finite representation type, grows linearly if A is of tame representation type, and grows quadratically if A is of wild representation type. Moreover, r A (n) allows us to construct invariants of algebras which are finer than the representation type.
Let G be a linear algebraic group over a field. We show that, under mild assumptions, in a family of primitive generically free G-varieties over a base variety B the essential dimension of the geometric fibers may drop on a countable union of Zariski closed subsets of B and stays constant away from this countable union. We give several applications of this result.
Jannsen asked whether the rational cycle class map in continuous
$\ell $
-adic cohomology is injective, in every codimension for all smooth projective varieties over a field of finite type over the prime field. As recently pointed out by Schreieder, the integral version of Jannsen’s question is also of interest. We exhibit several examples showing that the answer to the integral version is negative in general. Our examples also have consequences for the coniveau filtration on Chow groups and the transcendental Abel-Jacobi map constructed by Schreieder.
We determine the rational Picard group of the moduli spaces of smooth pointed hyperelliptic curves and of their Deligne-Mumford compactification, over the field of complex numbers.
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