Tensor product surfaces and graded syzygies

Technical Summary of "Tensor Product Surfaces and Graded Syzygies" by Matthew Weaver

Problem Statement
The paper addresses the implicitization problem for tensor product surfaces. Specifically, given a four-dimensional vector space $U \subseteq H^0(\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a, b))$ with a basis of bihomogeneous polynomials $\{p_0, p_1, p_2, p_3\}$, the goal is to determine the implicit equation of the rational map $\phi_U: \mathbb{P}^1 \times \mathbb{P}^1 \dashrightarrow \mathbb{P}^3$. The image of this map, denoted $X_U$, is a surface in $\mathbb{P}^3$. While standard elimination techniques (Gröbner bases, resultants) exist, they are often computationally expensive, particularly in the multigraded setting of $\mathbb{P}^1 \times \mathbb{P}^1$. The paper focuses on the case where the ideal $I_U = (p_0, p_1, p_2, p_3)$ admits a "singly graded" syzygy, a condition that has been shown in prior work to simplify the implicitization process.

Methodology
The author employs the approximation complex $\mathcal{Z}$, developed by Herzog, Simis, and Vasconcelos, which relates the Rees algebra of an ideal to its symmetric algebra. The core strategy relies on the fact that the implicit equation of $X_U$ can be recovered from the determinant of a specific matrix representation of the first differential $d_1$ of a strand of $\mathcal{Z}$, provided the strand is square and acyclic in a specific degree.

The methodology proceeds as follows:

1. Singly Graded Syzygy: The paper assumes $I_U$ possesses a minimal syzygy $S$ in a singly graded bidegree, specifically $(0, n)$ (or symmetrically $(n, 0)$), where $n \ge 1$.
2. Associated Subspace $V$: From the syzygy $S$, the author constructs a $K$-vector subspace $V \subseteq U$. If $S = \sum_{j=0}^n (\sum_{i=0}^3 a_{ij} p_i) u^{n-j}v^j$, then $V$ is spanned by the polynomials $f_j = \sum_{i=0}^3 a_{ij} p_i$.
3. Case Analysis by Dimension: The paper analyzes three distinct cases based on the dimension of $V$ (which is shown to be 2, 3, or 4):
   • Case $\dim V = 2$: The entries of $S$ form a regular sequence. The author constructs two additional syzygies ($S_1, S_2$) using the Hilbert-Burch structure of the ideal generated by these entries.
   • Case $\dim V = 3$: The entries of $S$ generate an Artinian ideal of codimension two. The author utilizes the Hilbert-Burch theorem to describe the syzygy matrix of these entries and constructs two additional syzygies ($S_2, S_3$) by lifting relations involving the remaining generator of $I_U$.
   • Case $\dim V = 4$: Here $V = U$, meaning the basis of $V$ generates the entire ideal $I_U$. The entries of $S$ generate an Artinian ideal with a more complex resolution. The author constructs three additional syzygies ($S_1, S_2, S_3$) using a hierarchy of Hilbert-Burch matrices and their associated syzygy matrices.
4. Matrix Construction: In each case, the author explicitly constructs a set of syzygies $\{S, S_1, S_2, S_3\}$ (or a subset thereof) that span a free module of rank sufficient to form a square matrix of size $2ab \times 2ab$ in the bidegree $(2a-1, b-1)$.

Key Contributions and Results
The primary contribution is a generalization of previous results by Duarte and Schenck (for linear syzygies, $n=1$) and the author's own previous work (for quadratic syzygies, $n=2$) to the general case of a singly graded syzygy in degree $n$.

• Theorem 4.4 ($\dim V = 2$): If $I_U$ has a minimal syzygy $S$ in bidegree $(0, n)$ with $b \ge 2n-1$ and $\dim V = 2$, then $I_U$ admits two additional syzygies $S_1, S_2$ in bidegree $(a, b-n)$ such that the space spanned by $\{S, S_1, S_2\}$ in bidegree $(2a-1, b-1)$ has dimension $2ab$.
• Theorem 5.11 ($\dim V = 3$): If $\dim V = 3$, there exist syzygies $S_1, S_2, S_3$ with specific bidegrees involving parameters $\mu$ derived from the resolution of the entries of $S$, such that the span of $\{S, S_1, S_2, S_3\}$ in the target bidegree has dimension $2ab$.
• Theorem 6.8 ($\dim V = 4$): If $\dim V = 4$, there exist syzygies $S_1, S_2, S_3$ with bidegrees determined by parameters $\mu_1, \mu_2$ from the resolution of the entries of $S$, such that the span of $\{S, S_1, S_2, S_3\}$ has dimension $2ab$.

In all cases, the determinant of the $2ab \times 2ab$ matrix representation of the differential $d_1$ formed by these syzygies is a power of the implicit equation of $X_U$.

Significance and Claims
The paper claims that these results provide a more efficient computational method for finding implicit equations of tensor product surfaces compared to computing the full syzygy module $\text{syz}(I_U)$ over the four-variable ring $K[s, t, u, v]$.

• Efficiency: The constructed syzygies are explicit and formulaic. They rely on syzygies of ideals in the two-variable subring $K[u, v]$ (or $K[s, t]$), which are computationally cheaper to determine than the full module over the four variables.
• Generality: The work unifies and extends the cases $n=1$ and $n=2$ to arbitrary $n$, provided the condition $b \ge 2n-1$ holds.
• Implementation: The author notes that these constructions are readily implementable in computer algebra systems like Macaulay2. Computational examples suggest that for larger parameters, this method is faster than computing the full syzygy module.

The paper concludes by posing open questions regarding the extension of these methods to surfaces with basepoints, the uniqueness of minimal singly graded syzygies, and the geometric implications of these syzygies on the singular locus of the surface.