Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 113 / 5

The direct image of a coherent sheaf under a closed immersion puts $\mathcal{G}=f_{\ast }\mathcal{F}$ in the category of coherent sheaves on ${\mathbb{P}}_k^n$. By sheaf cohomology under a closed inclusion and the supplied compatibility of twisting with direct image,Here is a proof of the required Serre vanishing on projective space. A coherent sheaf on ${\mathbb{P}}_k^n$ is the sheaf associated with a graded module for a finite graded $k[t_0,\dots ,t_n]$-module. Equivalently, it has a presentation by finite sums of twisting sheaves. Use a finite twisting resolution of a coherent sheaf on projective space: resolve the graded module by a finite graded free resolution, using the Hilbert syzygy theorem, and sheafify; exactness of localization preserves the resolution. Its terms are finite sums of $\mathcal{O}(a)$.

Choose $d$ sufficiently large that all twists $a+d$ occurring in these finitely many terms are nonnegative. The cohomology of twisting sheaves on projective space then vanishes in every positive degree for every resolution term. In a short exact sequence $0\to {\mathcal{K}}_{j+1}\to {\mathcal{E}}_j\to {\mathcal{K}}_j\to 0$, with ${\mathcal{K}}_0=\mathcal{G}$, the long exact sequence in sheaf cohomology identifies $H^p({\mathcal{K}}_j(d))$ with $H^{p+1}({\mathcal{K}}_{j+1}(d))$ for $p>0$. Iterating to the final acyclic term provesThe standard graded-module description used here is given in Stacks Project, Section 30.15; the vanishing follows from the displayed finite-resolution argument.

Choose the rational point $x=[1:0:\cdots :0]$ and the coherent sheaf $\mathcal{F}=i_{\ast }k$ supported there. By skyscraper sheaf cohomology, $H^0(X,\mathcal{F})=k$.

Its dual of a sheaf ${\mathcal{F}}^{\vee }=\mathcal{H}om(\mathcal{F},{\mathcal{O}}_X)$ is zero. Away from $x$ its source is zero. At $x$, a homomorphism $k\to {\mathcal{O}}_{X,x}$ must send one to an element annihilated by the maximal ideal. Since $n>0$, that ideal contains a nonzero coordinate parameter; the local ring is an integral domain, so the image must vanish. This proves that every local homomorphism vanishes. ConsequentlyThere is no contradiction with Serre duality: the ordinary sheaf dual suffices for locally free sheaves, but general coherent sheaves require an Ext functor. This example illustrates failure of ordinary sheaf-dual Serre duality for a skyscraper sheaf.

Eliminate $t_3$ using the linear equation. The homogeneous coordinate ring becomesThe cubic is a nonzero element of a polynomial integral domain, hence a non-zero-divisor. It gives the exact sequence of graded modulesThus $\dim B_m=\left(\genfrac{}{}{0ex}{}{m+2}{2}\right)-\left(\genfrac{}{}{0ex}{}{m-1}{2}\right)$ for $m\ge 3$. Equivalently, use the Hilbert series $(1-z^3)/(1-z{)}^3$, or the sheafified exact sequence on ${\mathbb{P}}^2$. The Hilbert polynomial of the plane cubic isIts degree is three and its arithmetic genus is one, although the curve is singular: its affine equation near $[1:0:0]$ is the cusp $t_2^2+t_1^3=0$. The Hilbert polynomial records the arithmetic genus, rather than the genus of the normalization.