Explicit description of cohomology modules of a coherent sheaf on $\mathbb P^2$

Question

I am trying to understand the basics of sheaf cohomology, so I tried on a concrete example, but things are still very unclear to me...

Let $\mathscr F$ be the coherent subsheaf of $\mathscr O_{\mathbb P^2}$ whose global sections are homogeneous polynomials of $\mathbb C[x,y,z]$ vanishing at $2$ given points $\mathbf a,\mathbf b\in\mathbb P^2$. Is there a way to describe explicitely the cohomology modules $H^1(\mathbb P^2,\mathscr F)$ and $H^2(\mathbb P^2,\mathscr F)$?

Answer 1

Let $X=\{a,b\}\subset \mathbb P^n\; (n\geq 1)$ be a two-point subset of $n$-dimensional projective space and $\mathcal F=\mathcal I_X$ the ideal sheaf of functions vanishing at $a$ and $b$.
This means that for an open subset $U\subset \mathbb P^n$ a section $s\in F(U)=\mathcal I_X(U)\subset \mathcal O(U)$ is a section in $\mathcal O(U)$ vanishing on $U\cap X$ (= a set with $0,1$ or $2$ elements).
We then have an exact sequence of coherent sheaves on $\mathbb P^n$: $$ 0\to \mathcal I_X\to \mathcal O_X \to Sky_X\to 0 $$ where $Sky_X$ is the sky scraper sheaf with stalk $k$ on $X$ and $0$ elsewhere.
Taking the associated cohomology long exact sequence we get $$0\to H^0(\mathbb P^n,\mathcal I_X)=0\to H^0(\mathbb P^n,\mathcal O_X)=k \to H^0(\mathcal P^n,Sky_X)=k^2\to H^1(\mathbb P^n,\mathcal I_X)\to H^1(\mathbb P^n,\mathcal O_X)=0 $$ from which we immediately deduce the required dimension $\operatorname {dim} _k H^1(\mathbb P^n,\mathcal I_X)=1$

As a check, notice that for $n=1$ we have $\mathcal I_X\cong\mathcal O(-2)$ and the equality $\operatorname {dim} _k H^1(\mathbb P^1,\mathcal O(-2))=1$ is easy to see, say by Serre duality.
Notice also that $n=1$ is the only value of $n$ for which $\mathcal I_X$ is an invertible sheaf (= line bundle)

Edit
A trivial modification of the above calculation shows that for a finite subset $X_r \subset \mathbb P^n$ with $r$ elements we have $$\operatorname {dim} _k H^1(\mathbb P^n,\mathcal I_{X_r})=r-1$$

Answer 2

Here are some preliminary computations and a first attempt. Please let me know if you see mistakes. To simplify, let $\mathbf a=(1:0)$ and $\mathbf b=(0:1)$ in $\mathbb P^1$ over an algebraically closed field $k$ and $\mathscr F$ be the ideal sheaf of $\mathcal O_{\mathbb P^1}$ vanishing on $\mathbf a$ and $\mathbf b$.

I wish to compute the cohomology $H^1(\mathbb P^1,\mathscr F)$ via the Cech complex.

Let $\mathscr U=\{D_+(x),D_+(y)\}$ be the standard cover. Clearly, the group of $2$-cochains is trivial and therefore $Z^1(\mathscr U,\mathscr F)=C^1(\mathscr U,\mathscr F)$, which is isomorphic to the abelian group $k[x]$

Let us compute the $1$-coboundaries. A $1$-cochain $f$ is a $1$-coboundary if there exists a $0$-cochain $h$ with $f(D_+(x)\cap D_+(y))=h(D_+(y))_{|D_+(x)\cap D_+(y)} - h(D_+(x))_{|D_+(x)\cap D_+(y)}.$ Since $h$ is a $0$-cochain, we have that $h(D_+(x))$ is a polynomial in $k[y]$ vanishing at 0 and $h(D_+(y))$ is a polynomial in $k[x]$ vanishing at 0.

Taking into account the restriction maps, we obtain that $f$ is a $1$-coboundary iff $f(D_+(x)\cap D_+(y))$ is the restriction of a polynomial in $k[x]$ vanishing at $0$. Consequently, $B^1(\mathscr U,\mathscr F)$ is isomorphic to $x k[x]$ and therefore $H^1(\mathbb P^1,\mathscr F)=k[x]/xk[x]=k$.

Does this make sense?