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AshyK
AshyK
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Let $X$ be a threefold with Kodaira dimension 2 such that the Iitaka map $\Phi :X \to Y$ is not isotrivial. The generic fiber of $\Phi$ is an elliptic curve.

Q1. How many such threefolds exist, and how many explicit examples can be given?

Q2. Let $\rho : T_Y \to H^1(X_y, T_{X_y})$ be the Kodaira-Spencer map (the differential of the moduli map $\mu : Y \to \mathcal{M}$, where $\mathcal{M}$ is the moduli space of elliptic curves). Is $\rho$ generically injective?

Let $X$ be a threefold with Kodaira dimension 2 such that the Iitaka map $\Phi :X \to Y$ is not isotrivial. The generic fiber of $\Phi$ is an elliptic curve.

Q1. How many such threefolds exist, and how many explicit examples can be given?

Q2. Let $\rho : T_Y \to H^1(X_y, T_{X_y})$ be the Kodaira-Spencer map (the differential of the moduli map $\mu : Y \to \mathcal{M}$, where $\mathcal{M}$ is the moduli space of elliptic curves). Is $\rho$ generically injective?

Let $X$ be a threefold with Kodaira dimension 2 such that the Iitaka map $\Phi :X \to Y$ is not isotrivial. The generic fiber of $\Phi$ is an elliptic curve.

Q1. How many such threefolds exist, and how many explicit examples can be given?

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AshyK
AshyK
  • 137
  • 8

Threefolds with Kodaira dimension 2 and non-isotrivial Iitaka map

Let $X$ be a threefold with Kodaira dimension 2 such that the Iitaka map $\Phi :X \to Y$ is not isotrivial. The generic fiber of $\Phi$ is an elliptic curve.

Q1. How many such threefolds exist, and how many explicit examples can be given?

Q2. Let $\rho : T_Y \to H^1(X_y, T_{X_y})$ be the Kodaira-Spencer map (the differential of the moduli map $\mu : Y \to \mathcal{M}$, where $\mathcal{M}$ is the moduli space of elliptic curves). Is $\rho$ generically injective?