Modulus of a rational map into a commutative algebraic group

For a rational map $\phi: X \to G$ from a normal algebraic variety $X$ to a commutative algebraic group $G$, we define the modulus of $\phi$ as an effective divisor on $X$. We study the properties of the modulus. This work generalizes the known theories for curves to higher dimensional varieties.Comment: 17 page

Geometric class field theory with bounded ramification for surfaces

Let X be a smooth projective geometrically irreducible variety over a perfect field k and D an effective divisor on X. We consider a relative Chow group of modulus D, the Albanese variety of X of modulus D and the Abel-Jacobi map with modulus. For X a surface over a finite field we prove a Roitman theorem with modulus and obtain a reciprocity law and an existence theorem for abelian coverings of X with ramification bounded by D. The previous version of this paper would require a stronger version of Bloch's moving lemma, for which I could not find a reference. I thank Shuji Saito for pointing this out. In this version we follow a different strategy: the main tool is a skeleton theorem that relates relative Cartier divisors on X to compatible systems of relative Cartier divisors on curves in X.Comment: 73 page

Thick source alpha counting: the measurement of thorium

The document attached has been archived with permission from the publisher.H. Sjostrand and J.R. Prescotthttp://www.aber.ac.uk/ancient-tl

Generalized Albanese and its Dual

A generalization of the Albanese variety to the case of a singular projective variety X over an algebraically closed field k is given in [ESV], where H. Esnault, V. Srinivas and E. Viehweg constructed a universal regular quotient of the Chow group CH0(X)deg 0 of 0-cycles of degree 0 modulo rational equivalence. This is a smooth connected commutative algebraic group, universal for rational maps from X to smooth commutative algebraic groups which factor through a homomorphism of groups CH0(X)deg 0 ��! G(k). Suppose now that in addition k is of characteristic 0. Interpreting this algebraic group as a generalized 1-motive in the sense of Laumon [L], we may ask for the dual 1-motive. The intention of these notes was to describe the functor which is represented by the dual 1-motive. This forms the main result of this work. The notion of dual 1-motive allows to treat the problem in a more general way: We consider certain categories of rational maps from a projective variety to commutative algebraic groups (the category of rational maps factoring through CH0(X)deg 0 is a special case). A necessary and sufficient condition for the existence of an object of such a category satisfying the universal mapping property is given, as well as a construction of these universal objects via their dual 1-motives. In particular, this provides an independent proof of the existence and an explicit construction of the universal regular quotient for algebraically closed base field of characteristic 0

The philosopher as artist: Ludwig Wittgenstein seen through Edoardo Paolozzi

In this article I argue that the strong fascination that Wittgenstein has had for artists cannot be explained primarily by the content of his work, and in particular not by his sporadic observation on aesthetics, but rather by stylistic features of his work formal aspects of his writing. Edoardo Paolozzi’s testimony shows that artists often had a feeling of acquaintance or familiarity with the philosopher, which I think is due to stylistic features of his work, such as the colloquial tone in which Wittgenstein shares his observation with the reader, but also the lack of long-winded arguments or explanations. In the concluding part I suggest that we can read Wittgenstein’s artworks of a specific kind: as philosophical works of art