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Euge
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Let ##k## be a field, and let ##X## be a subscheme of ##\mathbb{P}_k^2## defined by a single homogeneous equation ##f(x_0, x_1, x_2) = 0## of degree ##d##. Show that $$\dim_k H^1(X, \mathcal{O}_X) = \frac{(d-1)(d-2)}{2}$$