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1. CJM 2013 (vol 66 pp. 505)
Hodge Theory of Cyclic Covers Branched over a Union of Hyperplanes Suppose that $Y$ is a cyclic cover of projective space branched over
a hyperplane arrangement $D$, and that $U$ is the complement of the
ramification locus in $Y$. The first theorem implies that the
Beilinson-Hodge conjecture holds for $U$ if certain multiplicities of
$D$ are coprime to the degree of the cover. For instance this applies
when $D$ is reduced with normal crossings. The second theorem shows
that when $D$ has normal crossings and the degree of the cover is a
prime number, the generalized Hodge conjecture holds for any toroidal
resolution of $Y$. The last section contains some partial extensions
to more general nonabelian covers.
Keywords:Hodge cycles, hyperplane arrangements Category:14C30 |
2. CJM 2013 (vol 66 pp. 1225)
Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with $\mu=2$ |
Minimal Generators of the Defining Ideal of the Rees Algebra Associated with a Rational Plane Parametrization with $\mu=2$ We exhibit a set of minimal generators of the defining ideal of the
Rees Algebra associated with the ideal of three bivariate homogeneous
polynomials parametrizing a proper rational curve in projective plane,
having a minimal syzygy of degree 2.
Keywords:Rees Algebras, rational plane curves, minimal generators Categories:13A30, 14H50 |
3. CJM 2011 (vol 63 pp. 1038)
Critical Points and Resonance of Hyperplane Arrangements If $\Phi_\lambda$ is a master function corresponding to a hyperplane arrangement
$\mathcal A$ and a collection of weights $\lambda$, we investigate the relationship
between the critical set of $\Phi_\lambda$, the variety defined by the vanishing
of the one-form $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, and the resonance of $\lambda$.
For arrangements satisfying certain conditions, we show that if $\lambda$ is
resonant in dimension $p$, then the critical set
of $\Phi_\lambda$ has codimension
at most $p$. These include all free arrangements and all rank $3$ arrangements.
Keywords:hyperplane arrangement, master function, resonant weights, critical set Categories:32S22, 55N25, 52C35 |
4. CJM 2005 (vol 57 pp. 416)
Approximating Flats by Periodic Flats in \\CAT(0) Square Complexes We investigate the problem of whether every immersed flat plane in a
nonpositively curved square complex is the limit of periodic flat
planes. Using a branched cover, we reduce the problem to the case of
$\V$-complexes. We solve the problem for malnormal and cyclonormal
$\V$-complexes. We also solve the problem for complete square
complexes using a different approach. We give an application towards
deciding whether the elements of fundamental groups of the spaces we
study have commuting powers. We note a connection between the flat
approximation problem and subgroup separability.
Keywords:CAT(0), periodic flat planes Categories:20F67, 20F06 |
5. CJM 2000 (vol 52 pp. 123)
An Algorithm for Fat Points on $\mathbf{P}^2 Let $F$ be a divisor on the blow-up $X$ of $\pr^2$ at $r$ general
points $p_1, \dots, p_r$ and let $L$ be the total transform of a
line on $\pr^2$. An approach is presented for reducing the
computation of the dimension of the cokernel of the natural map
$\mu_F \colon \Gamma \bigl( \CO_X(F) \bigr) \otimes \Gamma \bigl(
\CO_X(L) \bigr) \to \Gamma \bigl( \CO_X(F) \otimes \CO_X(L) \bigr)$
to the case that $F$ is ample. As an application, a formula for
the dimension of the cokernel of $\mu_F$ is obtained when $r = 7$,
completely solving the problem of determining the modules in
minimal free resolutions of fat point subschemes\break
$m_1 p_1 + \cdots + m_7 p_7 \subset \pr^2$. All results hold for
an arbitrary algebraically closed ground field~$k$.
Keywords:Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl group Categories:13P10, 14C99, 13D02, 13H15 |