The Syzygy Matrix and the Differential for Rational Curves in Projective Space

Rational curves are crucial in the study of birational geometry and the arithmetic of varieties. Analyzing the restriction of vector bundles to a rational curve unveils important geometric properties of both the vector bundles and the underlying base space. Let $C\xrightarrow{\phi}\mathbb{P}^{n}$ be a rational curve in projective space over a field $K$. By Grothendieck’s theorem on vector bundles on $\mathbb{P}^{1}$, the restriction of the tangent bundle of $\mathbb{P}^{n}$ to C splits as a direct sum of line bundles $T_{\mathbb{P}^{n}}|_{C}=\bigoplus_{i=1}^{n}\mathcal{O}_{\mathbb{P}^{1}}(a_{i})$. We say that the restriction of tangent bundle is balanced if $|a_{i}-a_{j}|\leq 1$ for every $i,j$.

The splitting type of the restriction of the tangent bundle and normal bundle of a rational curve is an extensively studied topic. In [6], Ramella proved that the restriction of the tangent bundle of $\mathbb{P}^{n}$ to a general rational curve is balanced. In [5], Mandal studied the locus of morphisms from a rational curve to a Grassmannian with a specified splitting type of the restricted tangent bundle. In [7], Sacchiero proves that a general non-degenerate rational curve of degree $d$ in $\mathbb{P}^{n}$ has a balanced normal bundle when $n\geq 3$ and $char(K)=0$. In [4], Larson and Vogt extend this result to the case $char(K)=2$ . In [1], Coskun, Larson and Vogt proved that $N_{C/Gr(a,a+b)}$ is 2-balanced for a general rational curve in Grassmannian.

Let $C$ be a general rational curve in $\mathbb{P}^{n}$ of degree $d\geq n+1$. Then $T_{\mathbb{P}^{n}}|_{C}$ has degree $(n+1)d$. By Ramella’s result, we have a balanced restriction

$$T_{\mathbb{P}^{n}}|_{C}=\mathcal{O}_{\mathbb{P}^{1}}(d+q)^{\oplus a}\bigoplus\mathcal{O}_{\mathbb{P}^{1}}(d+q+1)^{\oplus b},$$

where $q=\lfloor\frac{d}{n}\rfloor$, $a=(q+1)n-d$ and $b=d-nq$. If we compose the tangent map $T_{C}\xrightarrow{d\phi}T_{\mathbb{P}^{n}}$ with the restriction map $T_{\mathbb{P}^{n}}\rightarrow T_{\mathbb{P}^{n}}|_{C}$, we get a morphism

$$f:\mathcal{O}_{\mathbb{P}^{1}}(2)\rightarrow\mathcal{O}_{\mathbb{P}^{1}}(d+q)^{\oplus a}\bigoplus\mathcal{O}_{\mathbb{P}^{1}}(d+q+1)^{\oplus b}.$$

Conversely, Eric Larson raised the following question. Given a morphism

$$f:\mathcal{O}_{\mathbb{P}^{1}}(2)\rightarrow\mathcal{O}_{\mathbb{P}^{1}}(d+q)^{\oplus a}\bigoplus\mathcal{O}_{\mathbb{P}^{1}}(d+q+1)^{\oplus b}$$

where $q$, $a$ and $b$ are defined above, when can we find a rational curve $C\subset\mathbb{P}^{n}$ such that $f$ is given by the composition of tangent map with restriction map?

When $d<n$, the curve $C$ is degenerate and not all morphisms $f$ can be represented by $C$. The restriction is not balanced except when $d=1$.

When $d=n$, $C$ is a rational normal curve. $Hom(\mathcal{O}_{\mathbb{P}^{1}}(2),\mathcal{O}_{\mathbb{P}^{1}}(n+1)^{n})\cong K^{n^{2}}$. Therefore, $f$ is induced by the restriction of the tangent map.

When $d\geq n+1$, the question is still open. Now we state our main conjecture and results on this problem.

As the main result of this paper, we provide computer assisted proofs of this conjecture for some numbers of $n$ and $d$.

Organization of the paper. In Section 2, we recall the preliminary facts needed in the rest of the paper. In Section 3 we explicitly write down the relation between $f$ and syzygy matrix of the rational curve. In Section 4, we show a computer assisted proof of Theorem 1.2 and Theorem 1.3.

Acknowledgements. I extend my deepest gratitude to my advisor, Izzet Coskun, for providing invaluable guidance and insightful suggestions throughout the course of my study at UIC. I would like to thank Ning Jiang for her constructive remarks on the proof of Theorem 1.3. I would also thank Yeqin Liu for many useful discussions.

In this sections, we collect necessary facts on rational curves and vector bundles on $\mathbb{P}^{n}$.

A rational curve is a curve which is birationally equivalent to $\mathbb{P}^{1}$. A rational curve $C\subset\mathbb{P}^{n}$ can be described as a map $[s:t]\mapsto[G_{0}(s,t),...,G_{n}(s,t)]$ where $G_{i}$ are homogeneous polynomials in variable $s,t$ of the same degree $d$ with no common zeros. Here $d$ is called the degree of the rational curve $C$.

Given a homogeneous ideal $I=(f_{1},...,f_{m})\subset K[x_{0},...,x_{n}]$. The first syzygy module $Syz(f_{1},...,f_{m})$ consists of $m$-tuples $(g_{1},...g_{m})$ of elements of $K[x_{0},...,x_{n}]$ such that $f_{1}g_{1}+,,,+f_{m}g_{m}=0$. The syzygy module can be presented as the kernel of a map $K[x_{0},...,x_{n}]^{m}\to I$. To describe the syzygy module, we often use a syzygy matrix. If we have relations $\sum_{j=1}^{m}l_{ij}f_{j}=0$ for $i=1,2,\ldots,t$, then these relations can be organized into a matrix $L$ such that

$$L\cdot\begin{pmatrix}f_{1}\\ f_{2}\\ \vdots\\ f_{m}\end{pmatrix}=0.$$

Here, $L$ is a $t\times m$ matrix with entries in $K[x_{0},...,x_{n}]$.

By the Birkhoff–Grothendieck theorem, every vector bundle $V$ of rank $r$ on $\mathbb{P}^{1}$ decomposes as a direct sum of line bundles

$$\mathcal{E}\cong\bigoplus_{i=1}^{r}\mathcal{O}_{\mathbb{P}^{1}}(a_{i})$$

for a unique sequence of integers $a_{1}\leq...\leq a_{r}$. The sequence of integers $a_{1},...,a_{r}$ is called the splitting type of $V$. We say that the vector bundle is $j$-balanced if $a_{1}-a_{r}\leq j$. If $j=0$, we say that $V$ is perfectly balanced. If $j=1$, we say that $V$ is balanced.

In [6], Ramella proved that the restriction of the tangent bundle of $\mathbb{P}^{n}$ to a general rational curve is balanced. Let $C$ be a general rational curve in $\mathbb{P}^{n}$ of degree $d\geq n+1$. Then $T_{\mathbb{P}^{n}}|_{C}$ has degree $(n+1)d$. We have a balanced restriction

$$T_{\mathbb{P}^{n}}|_{C}=\mathcal{O}_{\mathbb{P}^{1}}(d+q)^{\oplus a}\bigoplus\mathcal{O}_{\mathbb{P}^{1}}(d+q+1)^{\oplus b},$$

where $q=\lfloor\frac{d}{n}\rfloor$, $a=(q+1)n-d$ and $b=d-nq$.

In this part, we use the Euler sequence of projective space to set up a commutative diagram. This commutative diagram relates the syzygy matrix of the rational curve $C$ and the morphism $f$ in Conjecture 1.1 and Conjecture 1.2. We work over a field $K$ of general characteristic.

Let $n\geq 2$ be an integer. Consider the Euler sequence of projective space $\mathbb{P}^{n}$.

$$0\to\mathcal{O}_{\mathbb{P}^{n}}\to\mathcal{O}_{\mathbb{P}^{n}}(1)^{\oplus(n+1)}\to T_{\mathbb{P}^{n}}\to 0.$$

Let $C\subset\mathbb{P}^{n}$ be a rational curve of degree $d$. Restricting the Euler sequence to $C$, we get

$$0\to\mathcal{O}_{\mathbb{P}^{1}}\to\mathcal{O}_{\mathbb{P}^{1}}(d)^{\oplus(n+1)}\to T_{\mathbb{P}^{n}}|_{C}\to 0.$$

Suppose $C$ is a general curve, by Ramella’s result, we have

$$0\to\mathcal{O}_{\mathbb{P}^{1}}\to\mathcal{O}_{\mathbb{P}^{1}}(d)^{\oplus(n+1)}\to\mathcal{O}_{\mathbb{P}^{1}}(d+q)^{\oplus a}\bigoplus\mathcal{O}_{\mathbb{P}^{1}}(d+q+1)^{\oplus b}\to 0$$

where $q=\lfloor\frac{d}{n}\rfloor$, $a=(q+1)n-d$, $b=d-nq$. Combining this sequence with the Euler sequence of $\mathbb{P}^{1}$, we have the following commutative diagram.

$$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>\mathcal{O}_{\mathbb{P}^{1}}@>{}>{}>\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus 2}@>{\begin{pmatrix}-t&s\end{pmatrix}}>{}>\mathcal{O}_{\mathbb{P}^{1}}(2)@>{}>{}>0\\ @V{J}V{}V@V{FH}V{}V\\ 0@>{}>{}>\mathcal{O}_{\mathbb{P}^{1}}@>{G}>{}>\mathcal{O}_{\mathbb{P}^{1}}(d)^{\oplus(n+1)}@>{LP}>{}>\mathcal{O}_{\mathbb{P}^{1}}(d+q)^{\oplus a}\bigoplus\mathcal{O}_{\mathbb{P}^{1}}(d+q+1)^{\oplus b}@>{}>{}>0\\ \end{CD}$$

Let $s$, $t$ be the coordinate variables of $\mathbb{P}^{1}$. Morphisms in this commutative diagram can be represented by matrices of homogeneous polynomials with variables $s$, $t$.

$G=(G_{0},...,G_{n})$ is defined by the embedding of the rational curve $C\subset\mathbb{P}^{n}$ where

$$G_{i}=\sum_{j=0}^{d}g_{ij}s^{j}t^{d-j}$$

and $J$ is the Jacobian matrix of $G$.
$LP=\begin{pmatrix}L\\ P\end{pmatrix}$ is an $n\times(n+1)$ matrix representing the syzygy of the ideal generated by $G_{1},...,G_{n}$. $L=\begin{pmatrix}L_{10}&L_{01}&\cdots&L_{0n}\\ L_{20}&L_{11}&\cdots&L_{1n}\\ \vdots&\vdots&\ddots&\vdots\\ L_{a0}&L_{a1}&\cdots&L_{an}\end{pmatrix}$ where $L_{ki}=\sum_{j=0}^{q}l_{kij}s^{j}t^{q-j}$ for $k=1,...,a$ and $i=0,...,n$.
$P=\begin{pmatrix}P_{10}&P_{01}&\cdots&P_{0n}\\ P_{20}&P_{11}&\cdots&P_{1n}\\ \vdots&\vdots&\ddots&\vdots\\ P_{b0}&P_{b1}&\cdots&P_{bn}\end{pmatrix}$ where $P_{ki}=\sum_{j=0}^{q+1}p_{kij}s^{j}t^{q+1-j}$ for $k=1,...,b$ and $i=0,...,n$.
$FH=(F|H)^{\intercal}$ is the matrix representing the morphism $f$ in Conjecture 1.1 and Conjecture 1.2.
$F=(F_{1},...,F_{a})$ where $F_{i}=\sum_{j=0}^{d+q-2}f_{ij}s^{j}t^{d+q-2-j}$. $H=(H_{1},...,H_{b})$ where $H_{i}=\sum_{j=0}^{d+q-1}h_{ij}s^{j}t^{d+q-1-j}$. Since this is a commutative diagram, we get the equation of matrices

$$LP.J=FH.\begin{pmatrix}-t&s\end{pmatrix}.$$

(1)

Both sides of the equation (1) are $n\times 2$ matrices of homogeneous polynomials. If we can solve this system of equations, we will have a polynomial relation between the curve $C$ and the morphism $f$.

One natural idea is that we can represent the syzygy matrix $LP$ and the Jacobian matrix $J$ by $g_{ij}$ and solve (1) for $f_{ij}$ and $h_{ij}$ by $g_{ij}$. However, this computation is extremely difficult since it involves solving high degree polynomial equations with a huge number of variables.

Instead, we choose to fix the syzygy matrix $LP$ and construct a rational curve whose ideal has this syzygy matrix. Formally speaking, we take $G_{0},...,G_{n}$ to be $n\times n$-minors of $LP$. Then we compute the Jacobian $J$ and compare the coefficients of the polynomial in both sides of equation (1). It becomes a system of equations with variables $f_{ij}$, $h_{ij}$, $l_{kij}$ and $p_{kij}$.

The right hand side of 1 is

$$FH.\begin{pmatrix}-t&s\end{pmatrix}=\begin{pmatrix}-tF_{1}&sF_{1}\\ \vdots&\vdots\\ -tF_{a}&sF_{a}\\ -tH_{1}&sH_{1}\\ \vdots&\vdots\\ -tH_{b}&sH_{b}\\ \end{pmatrix}$$

Therefore, we can linearly solve for $f_{ij}$ and $h_{ij}$ by $l_{kij}$ and $p_{kij}$. The number of variables $f_{ij}$ and $h_{ij}$ is $a(d+q-1)+b(d+q)$. The number of variables $l_{kij}$ and $p_{kij}$ is $((q+1)a+(q+2)b)(n+1)$. The difference is $n^{2}+2n$. This shows we have enough variables $l_{kij}$ and $p_{kij}$ to represent $f_{ij}$ and $h_{ij}$.

To conclude, we have the following theorems.

We will analyze the surjectivity of $\Psi$ in next section. In the following example, we use a given syzygy matrix to compute the map $FH$.

In general, for each given pair of $n,d$, we can use Mathematica to symbolically compute $f_{ij}$ and $h_{ij}$. Here is the result for $n=2$ $d=4$.

Now, we use Mathematica computation to prove Theorem 1.2. We restate it for convenience.

Proof: By Theorem 3.1, we need to prove $\Psi$ is dominant. Since dominance is an open condition, it suffices to show the Jacobian matrix of $\Psi$ is of full rank at a random point.

As showed in Section 3, $\Psi$ is represented by the matrix $FH$. We can use Mathematica to symbolically compute $FH$ by the syzygy matrix $LP$ of the rational curve. Taking the Jacobian $J^{\prime}$ of $FH$, we get the matrix representation of the differential $d\Phi$. Now it suffices to show $J^{\prime}$ has full rank at a point. We use the random number generator in Mathematica to create a random numerical matrix $LP$ and plug $l_{kij}$ and $p_{kij}$ in the Jacobian $J^{\prime}$. Finally, we use the built-in rank-checking function in Mathematica to compute the rank of $J^{\prime}$. The result shows that $J^{\prime}$ has full rank. Thus, $\Psi$ is surjective at an Zariski open set of $K^{((q+1)a+(q+2)b)(n+1)}$. Therefore, $\Psi$ is dominant. $\Box$

To further explain this proof, we show the computation in detail for $n=2$, $d=3$.