I am looking for examples of smooth projective varieties $(X,H)$ with $H$ a polarization on $X$, $\dim \mbox{Pic}^0(X)=0$, $\mbox{Pic}(X) \not= \mathbb{Z}$ satisfying the property: for any two line bundles $L_1, L_2$ on $X$, we have $L_1 \cong L_2$ if and only if they have the same Hilbert polynomial.
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$\begingroup$ What do you assume about polarization? $\endgroup$– SashaMay 14, 2018 at 15:53
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$\begingroup$ @Sasha You can fix a polarization. $\endgroup$– ChenMay 14, 2018 at 16:22
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$\begingroup$ So, the question is about examples of polarized varieties $(X,H)$, right? $\endgroup$– SashaMay 14, 2018 at 16:31
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$\begingroup$ @Sasha I have edited the question $\endgroup$– ChenMay 14, 2018 at 22:05
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1 Answer
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Here is a simple example. Let $X = P^1 \times Q^3$ and $H = O(1,1)$. The Hilbert polynomial of the line bundle $O(a,b)$ is $$ P(t) = (t+a+1)(t+b+1)(t+b+2)(t+b+3/2)/3. $$ It has three integral roots $-a-1$, $-b-1$, $-b-2$, and one non-integral root $-b-3/2$. This allows to reconstruct $b$ from the Hilbert polynomial in a unique way, and after that also reconstruct $a$. Thus, the required property is satisfied.
