Some general facts on minimal free resolutions.

If C in P³ is a curve (=1-dimensional, equidimensional closed subscheme) its homogeneous ideal I(C) has a minimal free resolution as a graded S-module (S = k[x₀, ...,x₃]) which can be of one of the following two forms:

• 0 -> +S(-n_2i) -> +S(-n_1i) -> I(C) -> 0
• 0 -> +S(-n_3i)-> +S(-n_2i) -> +S(-n_1i) -> I(C) -> 0

(of course + is direct sum).

The first case occurs if and only if C is arithmetically Cohen-Macaulay (if C is smooth this is equivalent to being projectively normal). The simplest example of a.C.M. curves are complete intersections (if C is the complete intersection of F and G, the minimal free resolution of I(C) is given by the Koszul complex of F, G). In the sequel we will be interested with the second case which is the general one.

1. Definition: We set n₃₊= max {n_3i}, n_3-= min {n_3i} (similarly we have n₂₊, n₁₊, etc...). Also r_k will indicate the rank of +S(-n_ki).

2. Lemma: With notations as above:

1. r₃ + r₁ = r₂ + 1
2. n_3- > n_2- > n_1-
3. n₃₊ = c+4 (where c is the index of completeness of C: max{k/ h ¹(I_C(k)) > 0}), and #{i/ n_3i = n₃₊} = h¹(I_C(c)).
4. n₂₊ > n₁₊ and e+4 <= n_{2+ (where e is the index of speciality of C: max{k/ h²(IC(k)) > 0}).}
5. If n₃₊ <= n_{2+, then e >= c, e+4 = n2+ and +S(-n2i), where the sum is over the i such that n2i >= n3+, is a direct summand of the syzygies bundle.}

 

It is a good exercice to prove this lemma (otherwise see, for example, Ellia-Idà: "Some connections between equations and geometric propreties of curves in P³" in "Geometry and complex variables" Lect. Notes in Pure and Applied Math, 132 (M. Dekker)).