Primrose and Other Stories

Primrose and Other Stories is a short story collection that explores themes of family, loss, and legacy

Closed embeddings of C∗ in C2, part I

AbstractWe consider closed curves C≃C∗ in the affine plane S≃C2 that admit a good or very good asymptote. By this we mean a curve L≃C in S that in suitable coordinates for S is linear and tangent to C at infinity, and meets C once or not at all at finite distance. We classify these curves up to automorphism of S. Relying on the theory of open algebraic surfaces we first determine the possibilities for the singularities of C at infinity and then proceed to give explicit equations

Classification of singular Q-homology planes. II. C^1- and C*-rulings

A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We classify singular Q-homology planes which are C^1- or C*-ruled. We analyze their completions, the number of different rulings, the number of affine lines on it and we give constructions. Together with previously known results this completes the classification of Q-homology planes with smooth locus of non-general type. We show also that the dimension of a family of homeomorphic but non-isomorphic singular Q-homology planes having the same weighted boundary, singularities and Kodaira dimension can be arbitrarily big.Comment: 32 pages, to appear in Pacific J.

Classification of singular Q-homology planes. I. Structure and singularities

A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type. We show that if a Q-homology plane contains a non-quotient singularity then it is a quotient of an affine cone over a projective curve by an action of a finite group respecting the set of lines through the vertex. In particular, it is contractible, has negative Kodaira dimension and only one singular point. We describe minimal normal completions of such planes.Comment: improved results from Ph.D. thesis (University of Warsaw, 2009), 25 pages, to appear in Israel J. Mat

Affine modifications and affine hypersurfaces with a very transitive automorphism group

We study a kind of modification of an affine domain which produces another affine domain. First appeared in passing in the basic paper of O. Zariski (1942), it was further considered by E.D. Davis (1967). The first named author applied its geometric counterpart to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions which guarantee preservation of the topology under a modification. As an application, we show that the group of biregular automorphisms of the affine hypersurface $X \subset C^{k+2}$ given by the equation $uv=p(x_1,...,x_k)$ where $p \in C[x_1,...,x_k],$ acts $m-$transitively on the smooth part reg$X$ of $X$ for any $m \in N.$ We present examples of such hypersurfaces diffeomorphic to Euclidean spaces.Comment: 39 Pages, LaTeX; a revised version with minor changes and correction