Abstract. We construct a projective variety with discrete, non-finitely generated automorphism group. As an application, we show that there exists a complex projective variety with infinitely many non-isomorphic real forms.

We consider Dirichlet series ζg,α(s) = P ∞ n=1 g(nα)e −λns for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λn = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series P ∞ n=1 g(nα)z n . We prove that a Dirichlet series ζ(s) = P ∞ n=1 g(nα)/n s has an abscissa of convergence σ0 = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence is smaller or equal than 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ(s) has an analytic continuation to the entire complex plane.

We exhibit a pseudoeffective R-divisor D_\lambda on the blow-up of P^3 at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus B_-(D_\lambda) = \bigcup_{A ample}} B(D_\lambda+A) is not closed and that D_\lambda does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an R-divisor on the family of blow-ups of P^2 at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.Comment: Revamped introduction, various simplifications. To appear in Compositio Mathematica (with minor changes

Let X X be a smooth projective variety. The Iitaka dimension of a divisor D D is an important invariant, but it does not only depend on the numerical class of D D . However, there are several definitions of “numerical Iitaka dimension”, depending only on the numerical class. In this note, we show that there exists a pseuodoeffective R \mathbb {R} -divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective R \mathbb {R} -divisor D + D_+ for which h 0 ( X , ⌊ m D + ⌋ + A ) h^0(X,\left \lfloor {m D_+}\right \rfloor +A) is bounded above and below by multiples of m 3 / 2 m^{3/2} for any sufficiently ample A A .

A few questions about curves on surfaces. Rendiconti del circolo matematico di Palermo. Permanent WRAP URL:http://wrap.warwick.ac.uk/79423 Copyright and reuse:The Warwick Research Archive Portal (WRAP) makes this work by researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available.Copies of full items can be used for personal research or study, educational, or not-for profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. Publisher's statement:"The final publication is available at Springer via http://doi.org/10.1007/s12215-016-0284-4 A note on versions:The version presented here may differ from the published version or, version of record, if you wish to cite this item you are advised to consult the publisher's version. Please see the 'permanent WRAP url' above for details on accessing the published version and note that access may require a subscription. Abstract. In this note we address the following kind of question: let X be a smooth, irreducible, projective surface and D a divisor on X satisfying some sort of positivity hypothesis, then is there some multiple of D depending only on X which is effective or movable? We describe some examples, discuss some conjectures and prove some results that suggest that the answer should in general be negative, unless one puts some really strong hypotheses either on D or on X.