Curves with stable or semistable normal bundle on Fano hypersurfaces

Adding $1-g$ to both sides, it suffices to prove $\chi(B)/\text{rk}(B)\leq\lceil\chi(A)/\text{rk}(A)\rceil$. If $t\geq\chi(A)/\text{rk}(A)$ then $\chi(A(-D_{t}))\leq 0$, hence $H^{0}(A(-D_{t}))=0$. Therefore $H^{0}(B(-D_{t}))=0$, hence $\chi(B(-D_{t}))\leq 0$, i.e. $t\geq\chi(B)/\text{rk}(B)$. ∎

We will do the case $n=2m$ even, $n\geq 4$ as the case $n$ odd is similar and simpler (see comments at the end of the proof). Assume first that $e=2n-2$. Consider a fang degeneration

where

in which $E_{1}\subset P_{1},E_{2}\subset P_{2}$ are exceptional divisors and $P_{0}$ is constructed via isomorphisms $E_{1}\simeq E\simeq E_{2}$ which may be assume general. There is a standard smoothing of $P_{0}$ to $\mathbb{P}^{n}$. Consider curves

with each being a birational transform of a general rational curve of degree $e_{1}$ resp. $e_{2}$ meeting $\mathbb{P}^{m}$ resp. $\mathbb{P}^{m-1}$ twice, such that

and such that

Then $C_{0}=C_{1}\cup C_{2}$ is a nodal, lci ’fish’ curve in $P_{0}$ and smooths out to an elliptic curve $C_{*}$ of degree $e=e_{1}+e_{2}-2$ in $\mathbb{P}^{n}$ whose normal bundle is a deformation of $N_{C_{0}/P_{0}}=N_{C_{1}/P_{1}}\cup N_{C_{2}/P_{2}}$. Note that every degree $e$ is in Theorem 1 is obtained. Now using Lemma 31 of [14], we may assume each $C_{i}$ is balanced in $P_{i},i=1,2$ , so that

where

Now set $a=a_{1}+a_{2}$ and note that $N_{0}$ has slope

Because $N_{C_{k}/P_{k}}$ is balanced, it is easy to see that so is its general down modification.

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Note we have natural identifications

and likewise for $q$. The blowdown map $P_{1}\to\mathbb{P}^{n}$ contracts the vertical factor $\mathbb{P}^{m-1}$ of $E_{1}$ and because the upper subbundle $a_{1}\mathcal{O}(d_{1}+1)\subset N_{C_{1}/P_{1}}$ maps isomorphically to its image in $N_{C_{1}/\mathbb{P}^{n}}$, it follows that the fibre of the upper subbundle at $p$ is not contained in the vertical subspace $T_{p}\mathbb{P}^{m-1}\subset T_{p}E_{1}$, and likewise at $q$. Ditto for $N_{C_{2}/P_{2}}$.

Now I claim that with general choices, the upper subspaces of $N_{C_{1}/P_{1}}$ and $N_{C_{2}/P_{2}}$ at $p$ are in general position, and likewise for the exterior powers. To this end we use automorphisms. The automorphisms of $\mathbb{P}^{n}$ stabilizing $\mathbb{P}^{m}$ lift to automorphsms of $P_{1}$ that send $E_{1}$ to itself and are compatible with the projection $E_{1}\to\mathbb{P}^{m}$ (i.e. mapping a fibre to a fibre). Now the automorphism group of $P_{1}$ fixing $p,q$ maps surjectively to the automorphism group of $(E_{1}/\mathbb{P}^{m},p,q)$ and the latter acts transitively up to scalars on the ’nonvertical pairs’, i.e. pairs $(v_{p},v_{q})\in T_{p}E_{1}\oplus T_{q}E_{2}$ such that $v_{p}\not\in T_{p}\mathbb{P}^{m},v_{q}\not\in T_{q}\mathbb{P}^{m}$. Such automorphisms of $P_{1}$ move $C_{1}$ through $p,q$, compatibly with the isomorphism (1), hence also move the upper subsheaf $a\mathcal{O}(d_{1}+1)\subset N_{C_{1}/P_{1}}$ so its fibres at $p$ and $q$ are general subspaces. Ditto for $C_{2}$.

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Now the following Lemma concludes the proof of Theorem 1.

Now in case $n=2m+1$ odd the argument is the same using $P_{0}=P_{1}\cup P_{2}$ with $P_{i}=B_{\mathbb{P}^{m}}\mathbb{P}^{n},i=1,2$ glued along $E_{1}\simeq E_{2}\simeq\mathbb{P}^{m}\times\mathbb{P}^{m}$. ∎

We use induction on $g$, the case $g=1$ being contained in Theorem 1 We use the same fang degeneration of $\mathbb{P}^{n}$ as in Theorem 1 but this time we put $A$ on $P_{1}$. Again we consider a curve

For $C_{1}$ we use the birational transform in $P_{1}$ of a rational normal curve , with $C_{1}\cap E=\{p,q\}$. Thus we have a perfect normal bundle

For $C_{2}$ we however use a disjoint union

of curves of respective genera $1$, $g-1$ and respective degrees

(so that $\ell-6\geq 6(g-1)+2$), such that

meeting the common divisor $E$ and $C_{1}$ in $p$ resp. $q$. Thus $C_{0}$ has arithmetic genus $g$ and is smoothable to a curve of genus $g$ and degree $e=2m+1+e_{2,1}+e_{2,2}-2$ in $\mathbb{P}^{n}$. By induction , the normal bundles $N_{C_{2,i}}/P_{2}$ are semistable for $i=1,2$, as are their general down modifications. Therefore by the elementary Lemma below so is, in a suitable sense $N_{C_{0}/P_{0}}$ and likewise the smoothing $N_{C/P}$.

This completes the proof in case $n=2m+1$ odd. In case $n=2m$ even we let $P_{1}=B_{\mathbb{P}^{m-1}}\mathbb{P}^{n},P_{2}=B_{\mathbb{P}^{m}}\mathbb{P}^{n}$ and proceed similarly. ∎

We will use the same fan- quasi cone degeneration as in [13]. Thus we take

with $P_{1}$ the blowup pn $\mathbb{P}^{n+1}$ at a point $p$, with exceptional divisor $E=\mathbb{P}^{n}$, and $P_{2}=\mathbb{P}^{n+1}$ containing $E$ as a hyperplane. In $P_{0}$ we consider a Cartier divisor

with $X_{1}$ the blowup of a quasi-cone $\bar{X}_{1}$, i.e. a hypersurface of degree $n+1$ with multiplicity $n$ at $p$, and $X_{2}$ a general hyperurface of degree $n$ in $P_{2}=\mathbb{P}^{n+1}$. Then $X_{0}\subset P_{0}$ smooths out to a hypersurface $X\subset\mathbb{P}^{n+1}$ of degree $n+1$. As noted in [13], §4, $X_{1}$ may be realized as the blowup of $\mathbb{P}^{n}$ in an $(n,n+1)$ complete intersection

where $F_{n}$ corresponds to $X_{1}\cap E=X_{2}\cap E$ and $F_{n+1}$ (which is not unique) corresponds to a hyperplane section of $\bar{X}_{1}$ and its proper transform is $Z$. Proceeding as in [13], write

and consider a curve $C_{0}\subset X_{0}$ of the form

Here $C_{1}\subset X_{1}$ is the birational (=isomorphic) transform of a curve $C^{\prime}$ of genus $g$ and degree $e_{1}$ in $\mathbb{P}^{n}$, meeting $Y$ in $a$ points $p_{1},...,p_{a}$ with general tangents, whose normal bundle $N_{C^{\prime}/\mathbb{P}^{n}}$ is semistable and remains semistable after the general down modification at $p_{1},...,p_{a}$, corresponding to the tangent spaces $T_{p_{i}}Y$. Such a curve exists by Theorem 5 and meets $Z$ in $C^{\prime}\cap F_{n+1}\setminus\{p_{1},...,p_{a}\}$. The latter modification coincides with $N_{C_{1}/P_{1}}$. And as in [13], $C_{2}\subset X_{2}$ is a disjoint union of lines with trivial (hence semistable) normal bundle, meeting $C_{1}$ in $C_{1}\cap Z$. Now we have

Therefore e.g. by Lemma 7 above or by an argument as in [13], a smoothing $C\subset X$ of $C_{0}\subset X_{0}$ to a curve on a hypersurface of degree $n+1$ in $\mathbb{P}^{n+1}$ has semistable normal bundle.

The proof of c-hyper- semistability is the same, taking the modification centers to be general points on $C_{1}$. ∎

We will use the fang setup as in [13], §6. Thus we set $m=d-1$ and consider a limiting form of $\mathbb{P}^{n}$ which is a fang of the form

where

In $Z_{0}$ we consider a divisor which is a limiting form of a degree-$d$ hypersurface in $\mathbb{P}^{n}$ and has the form

where $X_{1}=\mathbb{P}_{\mathbb{P}^{m}}(G)\subset Z_{1}$ and $X_{2}\subset Z_{2}$ is fibred over $\mathbb{P}^{n-m-1}$ with general fibre a general hypersurface of degree $m$ in the $\mathbb{P}^{m+1}$ fibre of $Z_{2}$. We recall that $G$ is a rank-$n-m$ bundle over $\mathbb{P}^{m}$ which fits in an exact sequence

Then in $X_{0}$ we consider a connected lci curve of the form

where $C_{2}\subset X_{2}$ is a disjoint union of lines with trivial (hence c-semistable) normal bundle while $C_{1}\subset X_{1}$ is a suitable isomorphic lift of $\mathbb{P}^{n}$-degree $e$ of a smooth curve $C_{+}\subset\mathbb{P}^{m}$ of genus $g$ with c-semistable normal bundle. Then $C_{0}\subset X_{0}$ deforms to a smooth curve $C\subset X$ of degree $e$ and genus $g$ on a general hypersurface of degree $d$ and the normal bundle $N_{C/X}$ will be c-semistable if $N_{C_{0}/X_{0}}$ is, which in turn will be true provided $N_{C_{1}/X_{1}}$ is c-semistable. Thus it would suffice to show that with suitable choices $N_{C_{1}/X_{1}}$ may be assumed semistable. For convenience, let us call the $\mathbb{P}^{n}$ and $\mathbb{P}^{m}$ degrees of a curve $C_{1}\subset X_{1}$ the upper and lower degrees, say $e,e_{0}$, and the pair $(e,e_{0})$ the bidegree.

Our strategy for constructing $C_{1}$ with c-semistable normal bundle is based on the following exact sequence

combined with the following easy remark

The c-semistable case uses the fact that tensor products of semistable bundles are semistable in char. 0.

Now given $C_{+}\subset\mathbb{P}^{m}$, lifts of $C_{+}$ to $C_{1}\subset X_{1}=\mathbb{P}(G)$ correspond to invertible quotients

and the $\mathbb{P}^{n}$-degree of $C_{1}$ coincides with $\deg(B)$. The argument of [14], proof of Theorem 41 shows that if $e_{0}$ is large enough then such a lift $C_{1}\subset X_{1}$ exists for any $e:=\deg(B)\geq e_{0}$. Then we have as in loc. cit. $T_{v}=K^{*}(B)$ where $K$ is the kernel of the surjection $G|_{C_{+}}\to B$, and $T_{v}|_{C_{1}}$ has slope