# 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

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topDi 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

top- [1] E. Arbarello, M. Cornalba, P. A. Griffiths & J. Harris, Geometry of algebraic curves. Vol. I, Grund. Math. Wiss. 267, Springer, 1985. Zbl0559.14017
- [2] V. Di Gennaro & D. Franco, Factoriality and Néron-Severi groups, Commun. Contemp. Math.10 (2008), 745–764. Zbl1156.14006MR2446897
- [3] A. Dimca, Sheaves in topology, Universitext, Springer, 2004. Zbl1043.14003MR2050072
- [4] H. Flenner, L. O’Carroll & W. Vogel, Joins and intersections, Monographs in Mathematics, Springer, 1999. Zbl0939.14003MR1724388
- [5] W. Fulton, Intersection theory, Ergebnisse Math. Grenzg. 2, Springer, 1984. Zbl0541.14005MR732620
- [6] R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer, 1977. Zbl0367.14001MR463157
- [7] K. Lamotke, The topology of complex projective varieties after S. Lefschetz, Topology20 (1981), 15–51. Zbl0445.14010MR592569
- [8] E. J. N. Looijenga, Isolated singular points on complete intersections, London Mathematical Society Lecture Note Series 77, Cambridge University Press, 1984. Zbl0552.14002MR747303
- [9] A. Otwinowska & M. Saito, Monodromy of a family of hypersurfaces containing a given subvariety, Ann. Sci. École Norm. Sup.38 (2005), 365–386. Zbl1086.14010MR2166338
- [10] A. N. Parshin & I. R. Shafarevich (éds.), Algebraic geometry. III, Encyclopaedia of Mathematical Sciences 36, Springer, 1998. Zbl0886.14001MR1602371
- [11] E. H. Spanier, Algebraic topology, McGraw-Hill Book Co., 1966. Zbl0145.43303MR210112
- [12] J. H. M. Steenbrink, On the Picard group of certain smooth surfaces in weighted projective spaces, in Algebraic geometry (La Rábida, 1981), Lecture Notes in Math. 961, Springer, 1982, 302–313. Zbl0507.14025MR708341
- [13] C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés 10, Soc. Math. France, 2002. Zbl1032.14001MR1988456

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