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Ben C
Ben C
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Let $f : X \to D$ be a proper flat holomorphic family of complete algebraic curves over a disk and the central fiber is a nodal curve with a unique singular point $p \in X_0$. Suppose that every fiber is smooth except the central fiber. I am interested in the topology of $X$, specifically $\pi_1(X)$. How much information about the homotopy type of $X$ can be computed in terms of the smooth punctured fibration $f : X^* \to D^*$.

There is an exact sequence,

$$ 1 \to \pi_1(\Sigma_g) \to \pi_1(X^*) \to \mathbb{Z} \to 1 $$

where $g$ is the genus of the fiber. This presents the fundamental group as a semi-direct product,

$$ \pi_1(X) = \pi_1(\Sigma_g) \rtimes \mathbb{Z} $$

via the monodromy action. My guess is there is some way to compute $\pi_1(X)$ using ideas in the vein of vanishing cycles and Milnor fibers of the singularity.

In the case $g = 1$, I am guessing there is a unique topological type of $X$ given the monodromy. However, for $g > 1$ it is not clear if $f : X^* \to D^*$ contains enough information to determine $\pi_1(X)$.

Let $f : X \to D$ be a holomorphic family of complete algebraic curves over a disk and the central fiber is a nodal curve with a unique singular point $p \in X_0$. Suppose that every fiber is smooth except the central fiber. I am interested in the topology of $X$, specifically $\pi_1(X)$. How much information about the homotopy type of $X$ can be computed in terms of the smooth punctured fibration $f : X^* \to D^*$.

There is an exact sequence,

$$ 1 \to \pi_1(\Sigma_g) \to \pi_1(X^*) \to \mathbb{Z} \to 1 $$

where $g$ is the genus of the fiber. This presents the fundamental group as a semi-direct product,

$$ \pi_1(X) = \pi_1(\Sigma_g) \rtimes \mathbb{Z} $$

via the monodromy action. My guess is there is some way to compute $\pi_1(X)$ using ideas in the vein of vanishing cycles and Milnor fibers of the singularity.

In the case $g = 1$, I am guessing there is a unique topological type of $X$ given the monodromy. However, for $g > 1$ it is not clear if $f : X^* \to D^*$ contains enough information to determine $\pi_1(X)$.

Let $f : X \to D$ be a proper flat holomorphic family of complete algebraic curves over a disk and the central fiber is a nodal curve with a unique singular point $p \in X_0$. Suppose that every fiber is smooth except the central fiber. I am interested in the topology of $X$, specifically $\pi_1(X)$. How much information about the homotopy type of $X$ can be computed in terms of the smooth punctured fibration $f : X^* \to D^*$.

There is an exact sequence,

$$ 1 \to \pi_1(\Sigma_g) \to \pi_1(X^*) \to \mathbb{Z} \to 1 $$

where $g$ is the genus of the fiber. This presents the fundamental group as a semi-direct product,

$$ \pi_1(X) = \pi_1(\Sigma_g) \rtimes \mathbb{Z} $$

via the monodromy action. My guess is there is some way to compute $\pi_1(X)$ using ideas in the vein of vanishing cycles and Milnor fibers of the singularity.

In the case $g = 1$, I am guessing there is a unique topological type of $X$ given the monodromy. However, for $g > 1$ it is not clear if $f : X^* \to D^*$ contains enough information to determine $\pi_1(X)$.

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Ben C
Ben C
  • 3.6k
  • 7
  • 25

Topology of Lefschetz pencil

Let $f : X \to D$ be a holomorphic family of complete algebraic curves over a disk and the central fiber is a nodal curve with a unique singular point $p \in X_0$. Suppose that every fiber is smooth except the central fiber. I am interested in the topology of $X$, specifically $\pi_1(X)$. How much information about the homotopy type of $X$ can be computed in terms of the smooth punctured fibration $f : X^* \to D^*$.

There is an exact sequence,

$$ 1 \to \pi_1(\Sigma_g) \to \pi_1(X^*) \to \mathbb{Z} \to 1 $$

where $g$ is the genus of the fiber. This presents the fundamental group as a semi-direct product,

$$ \pi_1(X) = \pi_1(\Sigma_g) \rtimes \mathbb{Z} $$

via the monodromy action. My guess is there is some way to compute $\pi_1(X)$ using ideas in the vein of vanishing cycles and Milnor fibers of the singularity.

In the case $g = 1$, I am guessing there is a unique topological type of $X$ given the monodromy. However, for $g > 1$ it is not clear if $f : X^* \to D^*$ contains enough information to determine $\pi_1(X)$.