Monodromy of a family of hypersurfaces

TL;DR

This paper proves that the monodromy representation on a certain quotient of the middle cohomology of smooth hypersurfaces in a projective variety is irreducible, revealing fundamental symmetry properties of these hypersurfaces.

Contribution

It establishes the irreducibility of the monodromy representation on a specific cohomological quotient for a family of smooth hypersurfaces in a complex projective variety.

Findings

Monodromy representation is irreducible on the specified cohomological quotient.

The result applies to families of smooth divisors in a projective variety.

Provides insight into the symmetry and variation of hypersurfaces in algebraic geometry.

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 $d\geq \delta +1$, and assume the general divisor $X \in |H^0(Y,\ic_{Z,Y}(d))|$ is smooth. Denote by $H^m(X;\mathbb Q)_{\perp Z}^{\text{van}}$ 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 $H^m(X;\mathbb Q)_{\perp Z}^{\text{van}}$ for the family of smooth divisors $X \in |H^0(Y,\ic_{Z,Y}(d))|$ is irreducible.