Remarks on the normal bundles of generic rational curves

Abstract

In this note we give a different proof of Sacchiero’s theorem about the splitting type of the normal bundle of a generic rational curve. Moreover we discuss the existence and the construction of smooth monomial curves having generic type of the normal bundle.

Appendix

After this note was written, the paper [4] has appeared on ArXiv. Theorem 3.2 of [4] says that, for a monomial curve as in $(*),$

$$\begin{aligned} {\mathcal {N}}_{f}=\bigoplus \limits _{i=1}^{s-1}{\mathcal {O}}_{\mathbb {P} ^{1}}(d+h_{i-1}-h_{i+1}) \qquad (C-R \hbox { formula}). \end{aligned}$$

By using the above formula it is easy to prove the following theorem, saying exactly for which smooth monomial curves ${\mathcal {N}}_{f}$ splits according to Theorem 2.

Theorem 3

There exists a smooth monomial curve of degree d in $\mathbb {P} ^{s}, s\ge 3,$ with generic splitting type of ${\mathcal {N}}_{f}$ if and only if at least one of the following conditions are satisfied:

• (1) $d<3s/2.$

• (2) $d\ge 3s/2$ and $s-1$ is even.

• (3) $d\ge 3s/2$, $s-1$ is odd and $d=a(s/2)+b,$ with $0\le b\le (s/2)-a$ and $a\ge 3.$

Proof

• (1) It suffices to show that we can construct a smooth monomial curve with the generic splitting type for ${\mathcal {N}}_{f}$ we want. The condition $d<3s/2$ is equivalent to $d\ge 3e+4,$ hence the monomial curve is ”generic” in the sense of (I) and we have proved there that the splitting type of ${\mathcal {N}}_{f}$ is the generic one according to Theorem 2.

• (2) As for (1),  it suffices to show that we can construct a smooth monomial curve with the generic splitting type for ${\mathcal {N}}_{f}$ we want. We can argue as in (II).

• $(3)_{1}$ The condition is necessary. Let us assume that the sequence $h_{i-1}-h_{i+1}$ consists of at most two integers a and $a+1,$ with a appearing $q>0$ times. Let us consider the $h_{i}$ with even index, which are: $h_{0},h_{2},{\ldots },h_{s-2},h_{s},$ and the sum

  $$\begin{aligned} \sum \limits _{i=1}^{s/2}(h_{2(i-1)}-h_{2i})=h_{0}-h_{s}=d-0=d \end{aligned}$$

  Similarly, let us consider the $h_{i}$ with odd index: $h_{1},h_{3},{\ldots },h_{s-1}$ and the sum

  $$\begin{aligned} \sum \limits _{i=1}^{s/2-1}(h_{2i-1}-h_{2i+1})=h_{1}-h_{s-1}=(d-1)-1=d-2. \end{aligned}$$

  Suppose that within the even differences $h_{2(i-1)}-h_{2i}$ the number a appears $p_{1}$ times, and in the odd differences $h_{2i-1}-h_{2i+1}$ it appears $p_{2}$ times, and the number a appears, respectively, $q_{1}$ times and $q_{2}$ times with $p=p_{1}+p_{2}$ and $q=q_{1}+q_{2}$. Then one has the following relations, which we will call $(\clubsuit )$:

  $$\begin{aligned}&d=a(s/2)+p_{1}=q_{1}a+p_{1}(a+1),\qquad 0\le p_{1}\le s/2-1, p_{1}+q_{1}=s/2\\&d-2=a(s/2-1)+p_{2}=q_{2}a+p_{2}(a+1), 0\le p_{2}\le s/2-2, p_{2}+q_{2}=s/2-1. \end{aligned}$$

  From the first equality one also has that $d-2=a(s/2-1)+a+p_{1}-2,$ hence one obtains $p_{2}=a+p_{1}-2\le s/2-2,$ i.e. $p_{1}\le s/2-a.$ Putting $b=p_{1}$ we find the stated necessary condition: $d=a(s/2)+b$ with $0\le b\le s/2-a.$ Moreover we also have $s/2\ge a\ge 2s/(s-2),$ (recall that $s\ge 3$) hence $a\ge 3.$

• $(3)_{2}$ The condition is sufficient. We observe that, if $d=a(s/2)+b$ with $0\le b\le s/2-a,$ then one has $d-2=a(s/2-1)+a+b-2$ and one can put $p_{1}=b$ and $p_{2}=a+b-2\le s/2-2,$ so one can reproduce the relations $(\clubsuit ).$ Then one can choose the integers $h_{2i}$ with even indexes so that in the s / 2 differences $h_{2(i-1)}-h_{2i}$ the number a appears $q_{1}:=s/2-p_{1}$ and the number $a+1$ appears $p_{1}$ times. Similarly one can choose the integers $h_{2i-1}$ with odd indexes so that in the $s/2-1$ differences $h_{2i-1}-h_{2i+1}$ the number a appears $q_{2}:=s/2-1-p_{2}$ times and the number $a+1$ appears $p_{2}$ times. By applying the C-R formula it follows that the constructed sequence $h_{0},{\ldots },h_{s}$ defines a smooth monomial curve with the required splitting type for ${\mathcal {N}}_{f},$ because $h_{i-1}-h_{i+1}\in [a,a+1].$ $\square$