Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 113 / 3 / c / Solution

Let the degree-$d$ projective hypersurface $X_d$ be defined by the homogeneous polynomial $F$. Multiplication by $F$ identifies its ideal sheaf with ${\mathcal{O}}_{{\mathbb{P}}^n}(-d)$, giving the ideal-sheaf sequence of a projective hypersurfaceIn particular, the kernel in the question is ${\mathcal{O}}_{{\mathbb{P}}^n}(-d)$.

For $n=2$, additivity of the Euler characteristic in this short exact sequence and the line-bundle formulagiveHere $H^2(X_d,{\mathcal{O}}_{X_d})=0$ by the preceding cohomological-dimension argument. Moreover $H^0({\mathbb{P}}^2,\mathcal{O}(-d))=H^1({\mathbb{P}}^2,\mathcal{O}(-d))=0$ for $d>0$, so the long exact sequence gives $H^0(X_d,{\mathcal{O}}_{X_d})\cong k$. Consequentlywhich is the genus-degree formula for a projective plane curve.