Monodromy of a family of hypersurfaces

Vincenzo Di Gennaro; Davide Franco

Annales scientifiques de l'École Normale Supérieure (2009)

  • Volume: 42, Issue: 3, page 517-529
  • ISSN: 0012-9593

Abstract

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Let Y be an ( m + 1 ) -dimensional irreducible smooth complex projective variety embedded in a projective space. Let Z be a closed subscheme of Y , and δ be a positive integer such that ℐ Z , Y ( δ ) is generated by global sections. Fix an integer d ≥ δ + 1 , and assume the general divisor X ∈ | H 0 ( Y , ℐ Z , Y ( d ) ) | is smooth. Denote by H m ( X ; ℚ ) ⊥ Z van the quotient of H m ( X ; ℚ ) by the cohomology of Y and also by the cycle classes of the irreducible components of dimension m of Z . In the present paper we prove that the monodromy representation on H m ( X ; ℚ ) ⊥ Z van for the family of smooth divisors X ∈ | H 0 ( Y , ℐ Z , Y ( d ) ) | is irreducible.

How to cite

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Di Gennaro, Vincenzo, and Franco, Davide. "Monodromy of a family of hypersurfaces." Annales scientifiques de l'École Normale Supérieure 42.3 (2009): 517-529. <http://eudml.org/doc/272156>.

@article{DiGennaro2009,
abstract = {Let $Y$ be an $(m+1)$-dimensional irreducible smooth complex projective variety embedded in a projective space. Let $Z$ be a closed subscheme of $Y$, and $\delta $ be a positive integer such that $\mathcal \{I\}_\{Z,Y\}(\delta )$ is generated by global sections. Fix an integer $d\ge \delta +1$, and assume the general divisor $X \in |H^0(Y,\mathcal \{I\}_\{Z,Y\}(d))|$ is smooth. Denote by $H^m(X;\mathbb \{Q\})_\{\perp Z\}^\{\mathrm \{van\}\}$ the quotient of $H^m(X;\mathbb \{Q\})$ by the cohomology of $Y$ and also by the cycle classes of the irreducible components of dimension $m$ of $Z$. In the present paper we prove that the monodromy representation on $H^m(X;\mathbb \{Q\})_\{\perp Z\}^\{\mathrm \{van\}\}$ for the family of smooth divisors $X \in |H^0(Y,\mathcal \{I\}_\{Z,Y\}(d))|$ is irreducible.},
author = {Di Gennaro, Vincenzo, Franco, Davide},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {complex projective variety; linear system; Lefschetz theory; monodromy; isolated singularity; Milnor fibration},
language = {eng},
number = {3},
pages = {517-529},
publisher = {Société mathématique de France},
title = {Monodromy of a family of hypersurfaces},
url = {http://eudml.org/doc/272156},
volume = {42},
year = {2009},
}

TY - JOUR
AU - Di Gennaro, Vincenzo
AU - Franco, Davide
TI - Monodromy of a family of hypersurfaces
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2009
PB - Société mathématique de France
VL - 42
IS - 3
SP - 517
EP - 529
AB - Let $Y$ be an $(m+1)$-dimensional irreducible smooth complex projective variety embedded in a projective space. Let $Z$ be a closed subscheme of $Y$, and $\delta $ be a positive integer such that $\mathcal {I}_{Z,Y}(\delta )$ is generated by global sections. Fix an integer $d\ge \delta +1$, and assume the general divisor $X \in |H^0(Y,\mathcal {I}_{Z,Y}(d))|$ is smooth. Denote by $H^m(X;\mathbb {Q})_{\perp Z}^{\mathrm {van}}$ the quotient of $H^m(X;\mathbb {Q})$ by the cohomology of $Y$ and also by the cycle classes of the irreducible components of dimension $m$ of $Z$. In the present paper we prove that the monodromy representation on $H^m(X;\mathbb {Q})_{\perp Z}^{\mathrm {van}}$ for the family of smooth divisors $X \in |H^0(Y,\mathcal {I}_{Z,Y}(d))|$ is irreducible.
LA - eng
KW - complex projective variety; linear system; Lefschetz theory; monodromy; isolated singularity; Milnor fibration
UR - http://eudml.org/doc/272156
ER -

References

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