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1. CJM 2012 (vol 65 pp. 634)
Laplace Equations and the Weak Lefschetz Property We prove that $r$ independent homogeneous polynomials of the same degree $d$
become dependent when restricted to any hyperplane if and only if their inverse system parameterizes a variety
whose $(d-1)$-osculating spaces have dimension smaller than expected. This gives an equivalence
between an algebraic notion (called Weak Lefschetz Property)
and a differential geometric notion, concerning varieties which satisfy certain Laplace equations. In the toric case,
some relevant examples are classified and as byproduct we provide counterexamples to Ilardi's conjecture.
Keywords:osculating space, weak Lefschetz property, Laplace equations, toric threefold Categories:13E10, 14M25, 14N05, 14N15, 53A20 |
2. CJM 2010 (vol 62 pp. 1293)
Canonical Toric Fano Threefolds
An inductive approach to classifying all toric Fano varieties is
given. As an application of this technique, we present a
classification of the toric Fano threefolds with at worst canonical
singularities. Up to isomorphism, there are $674,\!688$ such
varieties.
Keywords:toric, Fano, threefold, canonical singularities, convex polytopes Categories:14J30, 14J30, 14M25, 52B20 |
3. CJM 2004 (vol 56 pp. 1094)
Cycle-Level Intersection Theory for Toric Varieties This paper addresses the problem of constructing a
cycle-level intersection theory for toric varieties.
We show that by making one global choice,
we can determine a cycle representative
for the intersection of an equivariant Cartier divisor with an invariant
cycle on a toric variety. For a toric variety
defined by a fan in $N$, the choice consists of giving an
inner product or a complete flag for $M_\Q=
\Qt \Hom(N,\mathbb{Z})$, or more
generally giving for each cone $\s$ in the fan a linear subspace of
$M_\Q$ complementary to $\s^\perp$, satisfying certain compatibility
conditions.
We show that these intersection cycles have properties analogous to the
usual intersections modulo rational equivalence.
If $X$ is simplicial (for instance, if $X$ is non-singular),
we obtain a commutative ring structure
to the invariant cycles of $X$ with rational
coefficients. This ring structure determines cycles representing
certain characteristic classes of the toric variety.
We also discuss
how to define intersection cycles that require no choices,
at the expense of increasing
the size of the coefficient field.
Keywords:toric varieties, intersection theory Categories:14M25, 14C17 |
4. CJM 2002 (vol 54 pp. 554)
Equivariant Embeddings into Smooth Toric Varieties We characterize embeddability of algebraic varieties into smooth toric
varieties and prevarieties. Our embedding results hold also in an
equivariant context and thus generalize a well-known embedding theorem
of Sumihiro on quasiprojective $G$-varieties. The main idea is to
reduce the embedding problem to the affine case. This is done by
constructing equivariant affine conoids, a tool which extends the
concept of an equivariant affine cone over a projective $G$-variety to
a more general framework.
Categories:14E25, 14C20, 14L30, 14M25 |
5. CJM 2000 (vol 52 pp. 348)
SingularitÃ©s quasi-ordinaires toriques et polyÃ¨dre de Newton du discriminant Nous \'etudions les polyn\^omes $F \in \C \{S_\tau\} [Y] $ \`a
coefficients dans l'anneau de germes de fonctions holomorphes au
point sp\'ecial d'une vari\'et\'e torique affine. Nous
g\'en\'eralisons \`a ce cas la param\'etrisation classique des
singularit\'es quasi-ordinaires. Cela fait intervenir d'une part
une g\'en\'eralization de l'algorithme de Newton-Puiseux, et
d'autre part une relation entre le poly\`edre de Newton du
discriminant de $F$ par rapport \`a $Y$ et celui de $F$ au moyen du
polytope-fibre de Billera et Sturmfels~\cite{Sturmfels}. Cela nous
permet enfin de calculer, sous des hypoth\`eses de non
d\'eg\'en\'erescence, les sommets du poly\`edre de Newton du
discriminant a partir de celui de $F$, et les coefficients
correspondants \`a partir des coefficients des exposants de $F$ qui
sont dans les ar\^etes de son poly\`edre de Newton.
Categories:14M25, 32S25 |