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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.

MSC:
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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