The Rees algebra of parametric curves via liftings. (English) Zbl 1423.13040
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Let \(\Bbbk\) be a field, \(R\) be the polynomial ring \(\Bbbk[T_0,T_1]\), and \(K\) be a height two ideal of \(R\) which is generated by the three homogeneous forms \(f_0,f_1,f_2\) of degree \(d\). The paper is about the defining ideal \(\mathfrak K\) of the Rees ring of \(K\). The authors would like to know a minimal generating set for \(\mathfrak K\), or the bi-degrees of a minimal generating set for \(\mathfrak K\), or all of the bi-homogeneous Betti numbers of \(\mathfrak K\). The questions are very hard and very important. Indeed, information about the singularities of the rational plane curve parameterized by \[ [f_0:f_1:f_2]:\mathbb P^1 \to \mathbb P^2 \] are encoded in the answers to the above questions. For example, the Rees algebra of \(K\) is the bi-homogeneous coordinate ring of the graph of the rational map which is given above.
Let \(Z_0,Z_1,Z_2\) and \(s\) be indeterminates. The ideal of interest, \(\mathfrak K\), is the kernel of the \(R\)-algebra homomorphism \[ \psi:R[Z_0,Z_1,Z_2]\to R[s], \] with \(\psi(Z_i)=f_is\). Everything in sight is bi-homogeneous where the bi-degrees are given by \begin{align*} \deg T_i&=(1,0), &\deg Z_i&=(0,1), &\deg s&=(-d,1),\\ \deg X&=(-\mu_1,1),& \deg Y&=(-\mu_2,1),&\deg X_i&=(0,1),\\ \deg (Y_i)&=(0,1),& \deg\alpha&=(d-\mu_1,0),\text{ and}& \deg \beta&=(d-\mu_2,0). \end{align*} The symbols \(X\), \(Y\), \(X_i\), and \(Y_i\) all represent indeterminates (that are defined below) and the symbols \(\alpha\) and \(\beta\) are homogeneous elements of \(R\) (that are also defined below).
More information about \(K\) must be identified before one can say anything about \(\mathfrak K\). Let \[ 0\to \begin{matrix} R(-d-d_1)\\ \oplus\\ R(-d-d_2)\end{matrix} \xrightarrow{\begin{bmatrix} p_0&q_0\\ p_1&q_1\\ p_2&q_2\end{bmatrix}} R(-d)^3\xrightarrow{\begin{bmatrix} f_0& f_1& f_2\end{bmatrix}} R \] be a minimal homogeneous resolution of \(R/K\). The Hilbert-Burch theorem guarantees that \(d_1+d_2=d\). We choose \(d_1\le d_2\). One argues that \((p_0,p_1,p_2)\) also is an ideal of height two in \(R\). Let \[ 0\to \begin{matrix} R(-d-\mu_1)\\ \oplus\\ R(-d-\mu_2)\end{matrix} \xrightarrow{\begin{bmatrix} A_0&B_0\\A_1&B_1\\A_2&B_2\end{bmatrix}} R(-d_1)^3\xrightarrow{\begin{bmatrix} p_0&p_1&p_2\end{bmatrix}} R \] be a minimal homogeneous resolution. Again, \(\mu_1+\mu_2=d_1\).
The vector \(\begin{bmatrix} f_0\\ f_1\\ f_2\end{bmatrix}\) is a relation on \(\begin{bmatrix} p_0& p_1& p_2\end{bmatrix}\); so there are homogeneous elements \(\alpha\) and \(\beta\) in \(R\) with \[ \begin{bmatrix} f_0\\ f_1\\ f_2\end{bmatrix} =\begin{bmatrix} A_0& B_0\\ A_1& B_1\\ A_2& B_2\end{bmatrix} \begin{bmatrix} \alpha\\ \beta\end{bmatrix}. \] Let \(A\) and \(B\) be matrices of constants with \[ \begin{bmatrix} A_0\\ A_1\\ A_2\end{bmatrix} =A\begin{bmatrix} T_0^{\mu_1}T_1^0\\ \vdots \\T_0^0T_1^{\mu_1}\end{bmatrix} \quad \text{and}\quad \begin{bmatrix} B_0\\ B_1\\ B_2\end{bmatrix} =B\begin{bmatrix} T_0^{\mu_2}T_1^0\\ \vdots \\ T_0^0T_1^{\mu_2}\end{bmatrix}. \] Let \(X_0,\dots,X_{\mu_1}\), \(Y_0,\dots,Y_{\mu_2}\), \(X\), and \(Y\) be indeterminates.
The main idea in the present paper is the observation that there is a commutative diagram of \(R\)-algebra homomorphisms of the form: \[ \begin{tikzcd} &R[\{X_i\},\{Y_i\}]\ar[d, "{\Phi'}"]\ar[rdd, "{\Phi}"]\\ R[Z_0,Z_1,Z_2]\ar[r, "{\Omega}"]\ar[ur, "{\Gamma}"]\ar[rrd, "{\psi}"] &R[X,Y]\ar[dr, "{\phi}"]\\ &&R[s], \end{tikzcd} \] where \[ \begin{bmatrix} \Gamma(Z_0)\\ \Gamma(Z_1) \\ \Gamma(Z_2)\end{bmatrix} = A\begin{bmatrix} X_0\\ \vdots\\ X_{\mu_1}\end{bmatrix} +B\begin{bmatrix} Y_0\\ \vdots\\ Y_{\mu_2}\end{bmatrix},\quad \quad\begin{bmatrix} \Omega(Z_0)\\ \Omega(Z_1) \\ \Omega(Z_2)\end{bmatrix} = \begin{bmatrix} A_0& B_0\\ A_1& B_1\\ A_2& B_2\end{bmatrix} \begin{bmatrix} X\\ Y\end{bmatrix}, \] \begin{align*}\Phi(X_i)&=\alpha T_0^{\mu_1-i}T_1^is,& \Phi(Y_i)&=\beta T_0^{\mu_2-i}T_1^is,&\phi(X)&=\alpha s,\\ \phi(Y)&=\beta s,&\Phi'(X_i)&=T_0^{\mu_1-i}T_1^iX,\text{ and}&\Phi'(Y_i)&=T_0^{\mu_2-i}T_1^iY. \end{align*} The goal is to calculate information about \(\ker \psi\). The kernel of \(\phi\) is well understood.
Let \(\Bbbk\) be a field, \(R\) be the polynomial ring \(\Bbbk[T_0,T_1]\), and \(K\) be a height two ideal of \(R\) which is generated by the three homogeneous forms \(f_0,f_1,f_2\) of degree \(d\). The paper is about the defining ideal \(\mathfrak K\) of the Rees ring of \(K\). The authors would like to know a minimal generating set for \(\mathfrak K\), or the bi-degrees of a minimal generating set for \(\mathfrak K\), or all of the bi-homogeneous Betti numbers of \(\mathfrak K\). The questions are very hard and very important. Indeed, information about the singularities of the rational plane curve parameterized by \[ [f_0:f_1:f_2]:\mathbb P^1 \to \mathbb P^2 \] are encoded in the answers to the above questions. For example, the Rees algebra of \(K\) is the bi-homogeneous coordinate ring of the graph of the rational map which is given above.
Let \(Z_0,Z_1,Z_2\) and \(s\) be indeterminates. The ideal of interest, \(\mathfrak K\), is the kernel of the \(R\)-algebra homomorphism \[ \psi:R[Z_0,Z_1,Z_2]\to R[s], \] with \(\psi(Z_i)=f_is\). Everything in sight is bi-homogeneous where the bi-degrees are given by \begin{align*} \deg T_i&=(1,0), &\deg Z_i&=(0,1), &\deg s&=(-d,1),\\ \deg X&=(-\mu_1,1),& \deg Y&=(-\mu_2,1),&\deg X_i&=(0,1),\\ \deg (Y_i)&=(0,1),& \deg\alpha&=(d-\mu_1,0),\text{ and}& \deg \beta&=(d-\mu_2,0). \end{align*} The symbols \(X\), \(Y\), \(X_i\), and \(Y_i\) all represent indeterminates (that are defined below) and the symbols \(\alpha\) and \(\beta\) are homogeneous elements of \(R\) (that are also defined below).
More information about \(K\) must be identified before one can say anything about \(\mathfrak K\). Let \[ 0\to \begin{matrix} R(-d-d_1)\\ \oplus\\ R(-d-d_2)\end{matrix} \xrightarrow{\begin{bmatrix} p_0&q_0\\ p_1&q_1\\ p_2&q_2\end{bmatrix}} R(-d)^3\xrightarrow{\begin{bmatrix} f_0& f_1& f_2\end{bmatrix}} R \] be a minimal homogeneous resolution of \(R/K\). The Hilbert-Burch theorem guarantees that \(d_1+d_2=d\). We choose \(d_1\le d_2\). One argues that \((p_0,p_1,p_2)\) also is an ideal of height two in \(R\). Let \[ 0\to \begin{matrix} R(-d-\mu_1)\\ \oplus\\ R(-d-\mu_2)\end{matrix} \xrightarrow{\begin{bmatrix} A_0&B_0\\A_1&B_1\\A_2&B_2\end{bmatrix}} R(-d_1)^3\xrightarrow{\begin{bmatrix} p_0&p_1&p_2\end{bmatrix}} R \] be a minimal homogeneous resolution. Again, \(\mu_1+\mu_2=d_1\).
The vector \(\begin{bmatrix} f_0\\ f_1\\ f_2\end{bmatrix}\) is a relation on \(\begin{bmatrix} p_0& p_1& p_2\end{bmatrix}\); so there are homogeneous elements \(\alpha\) and \(\beta\) in \(R\) with \[ \begin{bmatrix} f_0\\ f_1\\ f_2\end{bmatrix} =\begin{bmatrix} A_0& B_0\\ A_1& B_1\\ A_2& B_2\end{bmatrix} \begin{bmatrix} \alpha\\ \beta\end{bmatrix}. \] Let \(A\) and \(B\) be matrices of constants with \[ \begin{bmatrix} A_0\\ A_1\\ A_2\end{bmatrix} =A\begin{bmatrix} T_0^{\mu_1}T_1^0\\ \vdots \\T_0^0T_1^{\mu_1}\end{bmatrix} \quad \text{and}\quad \begin{bmatrix} B_0\\ B_1\\ B_2\end{bmatrix} =B\begin{bmatrix} T_0^{\mu_2}T_1^0\\ \vdots \\ T_0^0T_1^{\mu_2}\end{bmatrix}. \] Let \(X_0,\dots,X_{\mu_1}\), \(Y_0,\dots,Y_{\mu_2}\), \(X\), and \(Y\) be indeterminates.
The main idea in the present paper is the observation that there is a commutative diagram of \(R\)-algebra homomorphisms of the form: \[ \begin{tikzcd} &R[\{X_i\},\{Y_i\}]\ar[d, "{\Phi'}"]\ar[rdd, "{\Phi}"]\\ R[Z_0,Z_1,Z_2]\ar[r, "{\Omega}"]\ar[ur, "{\Gamma}"]\ar[rrd, "{\psi}"] &R[X,Y]\ar[dr, "{\phi}"]\\ &&R[s], \end{tikzcd} \] where \[ \begin{bmatrix} \Gamma(Z_0)\\ \Gamma(Z_1) \\ \Gamma(Z_2)\end{bmatrix} = A\begin{bmatrix} X_0\\ \vdots\\ X_{\mu_1}\end{bmatrix} +B\begin{bmatrix} Y_0\\ \vdots\\ Y_{\mu_2}\end{bmatrix},\quad \quad\begin{bmatrix} \Omega(Z_0)\\ \Omega(Z_1) \\ \Omega(Z_2)\end{bmatrix} = \begin{bmatrix} A_0& B_0\\ A_1& B_1\\ A_2& B_2\end{bmatrix} \begin{bmatrix} X\\ Y\end{bmatrix}, \] \begin{align*}\Phi(X_i)&=\alpha T_0^{\mu_1-i}T_1^is,& \Phi(Y_i)&=\beta T_0^{\mu_2-i}T_1^is,&\phi(X)&=\alpha s,\\ \phi(Y)&=\beta s,&\Phi'(X_i)&=T_0^{\mu_1-i}T_1^iX,\text{ and}&\Phi'(Y_i)&=T_0^{\mu_2-i}T_1^iY. \end{align*} The goal is to calculate information about \(\ker \psi\). The kernel of \(\phi\) is well understood.
Reviewer: Andrew Kustin (Columbia)
MSC:
| 13A30 | Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics |
| 14H50 | Plane and space curves |
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
Software:
Macaulay2References:
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