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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