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1. CJM 2013 (vol 66 pp. 505)

Arapura, Donu
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

2. CJM Online first

Cortadellas Benítez, Teresa; D'Andrea, Carlos
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)

Cohen, D.; Denham, G.; Falk, M.; Varchenko, A.
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)

Wise, Daniel T.
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)

Harbourne, Brian
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

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