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Exercise 111.45.6. In Situation 111.45.1 assume $n = 2$. Let $s_1, s_2, s_3 \in \Gamma (X, \mathcal{O}_ X(2))$ be three quadric equations. Consider the coherent sheaf

\[ \mathcal{F} = \mathop{\mathrm{Coker}}\left(\mathcal{O}_ X(-2)^{\oplus 3} \xrightarrow {s_1, s_2, s_3} \mathcal{O}_ X\right) \]

List the possible Hilbert polynomials of such $\mathcal{F}$. (Try to visualize intersections of quadrics in the projective plane.)

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View Exercise 111.45.6 as pdf