arXiv subjects

Hubert Flenner

Publications and source records attributed to Hubert Flenner.

23 records · Page 2Linked to original sources

arXiv2002

A Note on Generic Projections

Let $X \subseteq {\bf P}^N ={\bf P}^{2n}_K$ be a subvariety of dimension $n$ and $P \in {\bf P}^N$ a generic point. If the tangent variety Tan$ X$ is equal to ${\bf P}^N$ then for generic points $x$, $y$ of $X$ the projective tangent spaces $t_xX$ and $t_yX$ meet in one point $P=P(x,y)$. The main result of this paper is that the rational map $(x,y)\mapsto P(x,y)$ is dominant. In other words, a generic point $P$ is uniquely determined by the ramification locus $R(π_P)$ of the linear projection $π_P:X\to {\bf P}^{N-1}$.

math.AG↗

arXiv2001

Rational curves and rational singularities

We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a $\C^*$-action. For varieties with an isolated singularity, we show that the presence of sufficiently many rational curves outside the singular point strongly affects the character of the singularity. This provides an explanation of classical results due to H.A. Schwartz and G.H. Halphen on polynomial solutions of the generalized Fermat equation.

math.AG↗

arXiv2001

Contact Singularities

A contact singularity is a normal singularity $(V,0)$ together with a holomorphic contact form $η$ on $V\backslash$ Sing $V$ in a neighbourhood of 0, i.e. $η\wedge (dη)^r$ has no zero, where dim $V=2r+1$. The main result of this paper is that there are no isolated contact singularities.

math.AG↗

arXiv2000

Log-canonical forms and log canonical singularities

For a normal subvariety $V$ of ${\bf C}^n$ with a good ${\bf C}^*$-action we give a simple characterization for when it has only log canonical, log terminal or rational singularities. Moreover we are able to give formulas for the plurigenera of isolated singular points of such varieties and of the logarithmic Kodaira dimension of $V\backslash \{0\}$. For this purpose we introduce sheaves of $m$-canonical and $L^{2,m}$-canonical forms on normal complex spaces. For the case of affine varieties with good ${\bf C}^*$-action we give an explicit formula for these sheaves in terms of the grading of the dualizing sheaf and its tensor powers.

math.AG↗

arXiv1999

A Semiregularity Map for Modules and Applications to Deformations

This paper contains the details and complete proofs of our earlier announcement in math.AG/9907004 . We construct a general semiregularity map for algebraic cycles as asked for by S. Bloch in 1972. The existence of such a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture. Aside from generalizing and extending considerably previously known results in this direction, we give new applications to deformations of modules that encompass, for example, results of Artamkin and Mukai. The formation of the semiregularity map here involves powers of the cotangent complex, Atiyah classes, and trace maps, and is defined not only for subspaces of manifolds but for perfect complexes on arbitrary complex spaces. It generalizes in particular Illusie's treatment of the Chern character to the analytic context and specializes to Bloch's earlier description of the semiregularity map for locally complete intersections as well as to the infinitesimal Abel-Jacobi map for submanifolds.

math.AG↗