eBook ISBN:  9781470403492 
Product Code:  MEMO/159/756.E 
List Price:  $56.00 
MAA Member Price:  $50.40 
AMS Member Price:  $33.60 
eBook ISBN:  9781470403492 
Product Code:  MEMO/159/756.E 
List Price:  $56.00 
MAA Member Price:  $50.40 
AMS Member Price:  $33.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 159; 2002; 75 ppMSC: Primary 32; 14; 55;
In the present work we analyze two topological invariants of an embedding of an arrangement of rational plane curves in the projective complex plane, namely, the cohomology ring of the complement and the characteristic varieties. Our main result states that the cohomology ring of the complement to a rational arrangement is generated by logarithmic 1 and 2forms and its structure depends on a finite number of invariants of the curve (its combinatorial type). This theorem generalizes similar results for arrangements of lines by Brieskorn and OrlikSolomon. We also define a family of complexes (Generalized Aomoto Complexes) that play an important role in determining the characteristic varieties. They are equivalent to purely combinatorial complexes that were already defined for hyperplane arrangements (Aomoto Complexes). The resonance varieties of such complexes allow one to study the cohomology support loci of rank one local systems on the complement of a curve. In particular, we prove that the irreducible subgroups of the characteristic varieties of a rational arrangement are fully determined by its combinatorial data.
ReadershipGraduate students and research mathematicians interested in several complex variables, analytic spaces, algebraic geometry, and algebraic topology.

Table of Contents

Chapters

1. Preliminaries

2. Ring structure of $H^\bullet (\mathbb {P}^2 \mathcal {R}; \mathbb {C})$

3. Generalized Aomoto complexes

4. Characteristic varieties, local systems and rational arrangements

5. Examples


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In the present work we analyze two topological invariants of an embedding of an arrangement of rational plane curves in the projective complex plane, namely, the cohomology ring of the complement and the characteristic varieties. Our main result states that the cohomology ring of the complement to a rational arrangement is generated by logarithmic 1 and 2forms and its structure depends on a finite number of invariants of the curve (its combinatorial type). This theorem generalizes similar results for arrangements of lines by Brieskorn and OrlikSolomon. We also define a family of complexes (Generalized Aomoto Complexes) that play an important role in determining the characteristic varieties. They are equivalent to purely combinatorial complexes that were already defined for hyperplane arrangements (Aomoto Complexes). The resonance varieties of such complexes allow one to study the cohomology support loci of rank one local systems on the complement of a curve. In particular, we prove that the irreducible subgroups of the characteristic varieties of a rational arrangement are fully determined by its combinatorial data.
Graduate students and research mathematicians interested in several complex variables, analytic spaces, algebraic geometry, and algebraic topology.

Chapters

1. Preliminaries

2. Ring structure of $H^\bullet (\mathbb {P}^2 \mathcal {R}; \mathbb {C})$

3. Generalized Aomoto complexes

4. Characteristic varieties, local systems and rational arrangements

5. Examples