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No for the second --- even in the simplest case of a projective line, the polynomial $t^d + 1$$td + 1$ is the Hilbert polynomial (with respect to $L = O(d)$).
Yes for the first question, by Riemann--Roch.
No for the second --- even in the simplest case of a projective line, the polynomial $t^d + 1$ is the Hilbert polynomial (with respect to $L = O(d)$).
Yes for the first question, by Riemann--Roch.
No for the second --- even in the simplest case of a projective line, the polynomial $td + 1$ is the Hilbert polynomial (with respect to $L = O(d)$).
