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Let $X$ and $T$ be schemes smooth over $\mathbb{C}$ and suppose that $Pic(X \times T) \cong Pic(X) \times Pic(T)$.

Consider the following short exact sequence of locally free sheaves on $X$, $$ 0 \to \mathcal{L} \to \Omega_{X}^1 \to \mathcal{L}' \to 0 $$ where $\mathcal{L}$ and $\mathcal{L}'$are line bundles. In particular, $\mathcal{L}$ is a line subbundles of $\Omega_{X}^1$.

Let $p_S: X \times T \to X$.

Then we have a short exact sequence, $$ 0 \to p_s^*(\mathcal{L}) \to \Omega_{X \times T/T}^1 \to p_2^*(\mathcal{L}') \to 0 $$

Is it possible that there exist a line bundle $\mathcal{J}$ on $T$, such that

$$ 0 \to p_2^{*}(\mathcal{L}) \otimes p_1^*(\mathcal{J}) \to \Omega_{X \times T/T}^1 \to p_2^*(\mathcal{L}') \to 0?? $$

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Computing the determinant of $\Omega^1{X \times T/T}$ with the help of the last two sequences, one concludes that $p_1^*(\mathcal{J}) \cong \mathcal{O}_{X \times T}$, which implies $\mathcal{J} \cong \mathcal{O}_X$.

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  • $\begingroup$ What is the information that the determinant of $\Omega_{X \times T/T}$ provides? $\endgroup$ – user7090 Aug 10 '19 at 19:02

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