We first make some interesting generalizations to the previous post, problem 1. Recall that we showed that the blowup of $\PP{k}{2}$ at $n$ different points remains a (rational) Fano surface iff $n\le 8$ and the points are chosen in the projective plane such that no 3 are co-linear and no 6 lie on the same conic.
Now, consider smooth projective surface $S$ in say $\Bbb P_k^N$, $N\ge2$ that is a complete intersection of hypersurfaces. WLOG, $S$ is contained in no hyperplane. Let $\iota:S\ra \PP{k}{N}$ be its embedding into projective space, $\mathcal{L}:=\iota^*\OO_{\PP{k}{N}}(1)$, and write $S=H_1\cap…\cap H_{N-2}$ where the $H_i$ are hypersurfaces of degrees $d_i$. By the adjunction formula, for each $i$,
$$\omega_{H_i}=(\omega_{\PP{k}{N}}\otimes \OO_{\PP{k}{N}}(H_i))|_{H_i}=\OO_{H_i}(-N-1+d_i).$$
I claim that
$$\omega_S \cong \iota^*O_{\PP{k}{N}}(-N-1+\displaystyle\sum_{i=1}^{N-2}d_i).$$
In fact we will show the analogous statement for any smooth projective variety that is a complete intersection. We will proceed by induction on the number of hypersurfaces needed to define $S$. The base case is already done as it is the case for any one of the hypersurfaces. For each $1\le t\le N -3$, denote $S_t=H_1\cap…\cap H_t$ and assume that $\omega_{S_t}\cong\OO_{S_t}(-N-1+\displaystyle\sum_{i=1}^td_i)$. We see that $S=S_t\cap H_{N-3}$ is a divisor on $S_t$ by the projective dimension theorem and $S$ being a complete intersection. Applying the adjunction formula gives,
$\omega_S=(\omega_{S_t}\otimes\OO_{S_t}(S))|_S=\Big(\OO_{S_t}(-N-1+\displaystyle\sum_{i=1}^{N-3}d_i)\otimes\OO_{S_t}(H_{N-2}|_{S_t})\Big)\Big|_S$
$=\OO_S(-N-1+\displaystyle\sum_{i=1}^{N-2}d_i),$
which completes the induction. Set $D=\displaystyle\sum_{i=1}^{N-2}d_i$, and we have
$$\omega_S\cong\iota^*\OO_{\PP{k}{N}}(-N-1+D)$$.
Now, since pullbacks of ample line bundles under closed immersions are ample, we see that if $D<N+1$, then $-K_S$ is ample on $S$, i.e., $S$ is a Fano variety ($\OO_{\PP{k}{N}}(d)$ is ample iff very ample as it corresponds to the dth-degree Veronese embedding of $\PP{k}{N}$). Note that this result holds for any smooth complete intersection $S$. In particular, suppose $S$ is smooth projective m-fold that is a complete intersection of $N-m$ hypersurfaces. If $\displaystyle\sum_{i=1}^{N-m}d_i=:D_m<N+1$, then $S$ is a Fano variety. If we further require all the hypersurfaces to be at least degree 2, then for given $m$, we have
$$2N-2m\le D_m\le N+1\implies N\le 2m+1.$$
So, there are only finitely many $N$ that satisfy the above inequality, and each $N$ has only finitely many possible $D_m\le N$. Another way to say this is that for each $m\ge 2$ there are only finitely many distinct smooth, projective m-folds that are complete intersections of hypersurfaces of degrees at least 2 that are not canonically polarized ($K_S$ ample).
Finally, a corollary of our work above is,
Hartshorne exercise IV.3.3:
If $X$ is a curve of genus $\ge 2$ which is a complete intersection (II, Ex. 8.4) in some $\PP{k}{N}$, show that the canonical divisor $K$ is very ample. Conclude that a curve of genus 2 can never be a complete intersection in any $\PP{k}{N}$.
Indeed, given such $X$, $K_X\cong \OO_X(-N-1+\displaystyle\sum_{i=1}^{N-1}d_i)$. Because the embedding $\iota:X\ra \PP{k}{N}$ induces a homomorphism of Pic, we must have $\displaystyle\sum_{i=1}^{N-1}d_i>N+1$ because $\deg K_X>0$ so $K_X$ is not trivial. Then $K_X$ is always very ample, on a smooth projective curve $X$ of genus $\ge 2$, $K_X$ is very ample $\iff$ $X$ is not hyperelliptic. However, smooth projective curves of genus 2 are all hyperelliptic, ramified at 6 points, and the claim follows.
I’d like to make one last interesting note. Suppose $S\subset \Bbb P_{k}{3}$ is a smooth hypersurface of degree 3. Let $p\in S$ be any point and let $\varepsilon:\tilde S\ra S$ be the blowup at that point. By our previous work, we know,
$$\omega_S^*\cong\OO_{\PP{k}{3}}(4-3)|_S=\OO_S(1).$$
That is, $-K_S$ is the intersection of a general hyperplane in $\PP{k}{3}$ with $X$. By Bertini’s theorem, $-K_S$ is in fact a nonsingular curve on $S$. Then by the adjunction formula,
$$\omega_{-K_S}=(\omega_S\otimes\OO_{S}(-K_S))|_{-K_S}=\OO_{-K_S}.$$
Taking degrees gives $g(-K_S)=1$ so that $-K_S$ is a smooth elliptic curve.
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