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1. CJM 2011 (vol 64 pp. 123)
Gosset Polytopes in Picard Groups of del Pezzo Surfaces In this article, we study the correspondence between the geometry of
del Pezzo surfaces $S_{r}$ and the geometry of the $r$-dimensional Gosset
polytopes $(r-4)_{21}$. We construct Gosset polytopes $(r-4)_{21}$ in
$\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify
divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a-1)$-simplexes ($a\leq
r$), $(r-1)$-simplexes and $(r-1)$-crosspolytopes of the polytope $(r-4)_{21}$.
Then we explain how these classes correspond to skew $a$-lines($a\leq r$),
exceptional systems, and rulings, respectively.
As an application, we work on the monoidal transform for lines to study the
local geometry of the polytope $(r-4)_{21}$. And we show that the Gieser transformation
and the Bertini transformation induce a symmetry of polytopes $3_{21}$ and
$4_{21}$, respectively.
Categories:51M20, 14J26, 22E99 |
2. CJM 2008 (vol 60 pp. 64)
Classification of Linear Weighted Graphs Up to Blowing-Up and Blowing-Down We classify linear weighted graphs up to the
blowing-up and blowing-down operations which are relevant for the
study of algebraic surfaces.
Keywords:weighted graph, dual graph, blowing-up, algebraic surface Categories:14J26, 14E07, 14R05, 05C99 |
3. CJM 2007 (vol 59 pp. 1098)
Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions In this paper we study ruled surfaces which appear as an exceptional
surface in a succession of blowing-ups. In particular we prove
that the $e$-invariant of such a ruled exceptional surface $E$ is
strictly positive whenever its intersection with the other
exceptional surfaces does not contain a fiber (of $E$). This fact
immediately enables us to resolve an open problem concerning an
intersection configuration on such a ruled exceptional surface
consisting of three nonintersecting sections. In the second part
of the paper we apply the non-vanishing of $e$ to the study of the
poles of the well-known topological, Hodge and motivic zeta
functions.
Categories:14E15, 14J26, 14B05, 14J17, 32S45 |
4. CJM 2004 (vol 56 pp. 1145)
On Log $\mathbb Q$-Homology Planes and Weighted Projective Planes We classify normal affine surfaces with trivial Makar-Limanov
invariant and finite Picard group of the smooth locus, realizing them
as open subsets of weighted projective planes.
We also show that such a surface admits, up to conjugacy,
one or two $G_a$-actions.
Categories:14R05, 14J26, 14R20 |