I'm trying to show that if $X$ is a nonsingular curve of genus $g \geq 2$ which is a complete intersection $X = H_1 \cap H_2 \cap \dots \cap H_{n-1}$ in $\mathbb{P}^n$, then the canonical divisor $K$ on $X$ is very ample.
I've been able to solve this problem if I assume that the complete intersection is nonsingular, meaning that $H_1 \cap \dots \cap H_i$ is a nonsingular subvariety for each $1 \leq i \leq n-1$. To do this, I repeatedly applied the adjunction formula to show that if $\deg(H_i) = d_i$, then $\omega_X = \mathscr{O}_X(\sum_i d_i – n – 1)$, which is very ample when $g \geq 2$.
However, I'm not sure what to do when I don't assume nonsingularity of the complete intersection. I realize that it suffices to show that $\omega_X = \mathscr{O}_X(d)$ for some $d \in \mathbb{Z}$, but I'm not sure if this is even true in general. How can I handle the problem in this case?
Best Answer
I suspect that it might be possible to choose a sequence of hypersurfaces so that the partial intersections are always smooth, but I don't see how to do that right now. Here's an alternate approach involving the Koszul complex.
As $X\subset\Bbb P^n_k$ is projective over a field of codimension $n-1$, it has canonical bundle given by $\mathcal{E}xt^{n-1}(\mathcal{O}_X,\omega_{\Bbb P^n_k})$ (ref Hartshorne proposition III.7.5 or your favorite text on duality). Combining the facts that locally free sheaves are acyclic for sheaf Ext and $X$ is a complete intersection, we can compute a convenient resolution of $\mathcal{O}_X$ using the Koszul complex and then calculate sheaf Ext with that.
Supposing that each $H_i$ is cut out by the homogeneous polynomial $f_i$ of degree $d_i$, we have that $\mathcal{I}_X\subset\mathcal{O}_{\Bbb P^n_k}$ is the image of the map $\bigoplus_{i=1}^{n-1} \mathcal{O}_{\Bbb P^n_k}(-d_i)\to\mathcal{O}_X$ where the map from $\mathcal{O}_{\Bbb P^n_k}(-d_i)$ is multiplication by $f_i$. The Koszul complex is then $$0\to \bigwedge^{r-1}\left(\bigoplus_{i=1}^{n-1} \mathcal{O}_{\Bbb P^n_k}(-d_i)\right) \to \bigwedge^{r-2}\left(\bigoplus_{i=1}^{n-1} \mathcal{O}_{\Bbb P^n_k}(-d_i)\right)\to \cdots \to \bigoplus_{i=1}^{n-1} \mathcal{O}_{\Bbb P^n_k}(-d_i)\to \mathcal{O}_{\Bbb P^n_k}\to 0$$ where the $(r-1)^{th}$ wedge product is in degree $r-1$. As the $f_i$ form a regular sequence, this complex is exact and therefore forms a locally free resolution of $\mathcal{O}_X$ so we may use it to compute $\mathcal{E}xt(\mathcal{O}_X,\omega_{\Bbb P^n_k})$.
Simplifying our wedge products a bit, the degree $r-1$ term is $\mathcal{O}_{\Bbb P^n_k}(-\sum_{i=1}^{r-1} d_i)$, the degree $r-2$ term is $\bigoplus_{i=1}^{r-1} \mathcal{O}_{\Bbb P^n_k}(d_i - \sum_{j=1}^{r-1} d_j)$, and the map between them sends $1\mapsto (f_1,\cdots,f_{r-1})$. Applying $\mathcal{H}om(-,\omega_{\Bbb P^n})$, we find that the tail end of our complex is $$ \bigoplus_{i=1}^{r-1}\mathcal{O}_{\Bbb P^n_k}(-d_i-n-1+\sum_{j=1}^{r-1} d_j) \to \mathcal{O}_{\Bbb P^n_k}(\sum_{i=1}^{r-1} d_i-n-1)\to 0$$ where the nontrivial map sends the $i^{th}$ basis vector to $f_i$. But this means that this complex is just $\bigoplus_{i=1}^{n-1} \mathcal{O}_{\Bbb P^n_k}(-d_i)\to\mathcal{O}_X$ from earlier, twisted by $\sum d_i-n-1$, so $\omega_X\cong \mathcal{O}_X(\sum d_i-n-1)$ and we have the desired result without having to think about the nonsingularity of the intermediate intersections.