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Green’s conjecture: An orientation for algebraists. (English) Zbl 0792.14015
Free resolutions in commutative algebra and algebraic geometry, Proc. Conf., Sundance/UT (USA) 1990, Res. Notes Math. 2, 51-78 (1992).
[For the entire collection see Zbl 0745.00042.]
The first section of this paper leads to an algebraic conjecture generalizing Green’s conjecture.
Let \(S=k [x_ 0,\dots,x_ r]\), and let \(R=S/I\) be a homogeneous factor ring of \(S\). We assume that \(I\) contains no linear forms, and the projective dimension of \(S/I\) is \(m\). Then minimal free resolution \({\mathcal F}\) of \(S/I\) can be written as \[ \begin{aligned} 0 \leftarrow S/I & \leftarrow S \leftarrow S(-2)^{a_ 1} \oplus S(-3)^{b_ 1} \oplus \cdots \leftarrow S(-3)^{a_ 2} \oplus S(-4)^{b_ 2} \oplus \cdots\\ & \leftarrow \cdots \leftarrow S(-(m+1))^{a_ m} \oplus S(- (m+2))^{b_ m} \oplus S(-(m+3))^{c_ m} \oplus \cdots \leftarrow 0,\end{aligned} \] with \(a_ i,b_ i \in \mathbb{Z}\), \(a_ i,b_ i \geq 0\). We define the 2-linear strand of \({\mathcal F}\) to be the subcomplex \[ 0 \leftarrow S/I \leftarrow S \leftarrow S(-2)^{a_ 1} \leftarrow S(- 3)^{a_ 2} \leftarrow \cdots \leftarrow S(-(m+1))^{a_ m} \leftarrow 0. \] The author defines the length of the 2-linear strand to be the largest number \(n\) such that \(a_ n\neq 0\). He calls this \(n\) the 2-linear projective dimension and writes \(2\text{LP}(S/I)=n\).
If \(I\) is the ideal generated by the \(2 \times 2\) minors of a generic \(p \times q\) matrix, then the 2-linear strand is known to have length \(\geq p+q-3\). As a form of converse the author is lead to the following algebraic conjecture: Let \(k\) be an algebraically closed field of characteristic \(\neq 2\), and let \(I \subset S=k [x_ 0,\dots,x_ r]\) be a prime ideal, containing no linear form, whose quadratic part is spanned by quadrics of rank \(\leq 4\). If \(2\text{LP}(S/I)=n\), then \(I\) contains an ideal of \(2 \times 2\) minors of a 1-generic \(p \times q\) matrix with \(p+q-3=n\).
Green’s conjecture, from the algebraic point of view, is just the special case of this where (a) \(S/I\) is normal (= integrally closed); (b) \(\dim S/I=2\); (c) \(S/I\) is Gorenstein; (d) degree \(S/I = 2r\).
In section two the author considers the canonical ring of a non- hyperelliptic curve (= the homogeneous coordinate ring of the canonically embedded curve) and gets the geometric conjecture [M. L. Green, Invent. Math. 75, 85-104 (1984; Zbl 0542.14018)]:
The length of the 2-linear part of the resolution \({\mathcal F}\) of the canonical ring \(S/I\) of a curve of genus \(g\) and Clifford index \(c\) is \(2\text{LP}(S/I)=g-2-c\). The generic form there of becomes:
The free resolution of the canonical ring of a generic curve of genus \(g\) has \(a_{\lfloor g/2 \rfloor},\dots,a_{g-3}=0\), \(0,\dots,0\).
The third section of the paper surveys some approaches to Green’s conjecture.

14H99 Curves in algebraic geometry
13D25 Complexes (MSC2000)
13D05 Homological dimension and commutative rings
14H45 Special algebraic curves and curves of low genus
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