On Enriques-Babbage Theorem for singular curves

We propose a version of the Enriques-Babagge Theorem for a singular curve $C$, involving its canonical model $C’$. We provide a partial proof for an arbitrary curve $C$ and complete the proof for unicuspidal monomial curves by describing the generators of the ideal of $C’\subset\mathbb{P}^{g-1}$.

💡 Research Summary

The paper sets out to extend the classical Enriques‑Babbage theorem, which classifies smooth canonical curves as either cut out by quadrics, trigonal, or plane quintic, to the realm of singular (integral) curves. Starting from Max Noether’s theorem on projective normality of canonical curves, the authors reinterpret the theorem via Koszul cohomology and Clifford index, allowing torsion‑free sheaves to accommodate singularities.

A central construction is the Rosenlicht canonical model C′ ⊂ ℙ^{g‑1}, obtained by normalizing the curve C and mapping it using the global sections of the dualizing sheaf ω. The authors assume C′ is linearly normal, a condition that holds for all Gorenstein curves and for nearly Gorenstein curves (those with μ=1 in the notation of the paper). They introduce two numerical invariants at each point P: η_P = dim(ω_P / O_P) and μ_P = dim( bO_P / ω_P ), where bO denotes the blow‑up of the structure sheaf. A curve is called Kunz if η=1 (the minimal non‑Gorenstein case), and nearly Gorenstein if μ=1.

The first main result (Theorem 1) deals with arbitrary integral curves whose canonical model is linearly normal. For non‑Gorenstein curves it establishes: (i) K_{1,2}(C, ω)=0 ⇔ C′ is cut out by quadrics; (ii) C′ is always cut out by quadrics and cubics; (iii) If the curve is not Kunz, then quadrics alone suffice. These statements are proved by relating the Koszul cohomology of (C, ω) to that of (C′, O_{C′}(1)) and invoking the syzygy theorem.

The second part of Theorem 1 focuses on unicuspidal monomial curves. It shows that exactly one of the following holds: (i) C′ is cut out by quadrics; (ii) C is trigonal and Gorenstein; (iii) C is a plane quintic; (iv) C is Kunz (with a further refinement concerning the freedom of the g¹³). The proof combines a description of the scroll containing C′ (when trigonal) with the analysis of the ideal of C′.

The most technical contribution is Theorem 2, which gives an explicit set of generators for the homogeneous ideal I(C′) when C is a unicuspidal monomial curve that is nearly Gorenstein. Using the semigroup of values S of the singular point, the authors list the gaps G={ℓ₁,…,ℓ_g} and the Frobenius number γ. For each s∈{2,…,γ} they consider minimal partitions s = a_s + b_s and the associated binomials X_{a_s}X_{b_s} – X_{a_{s,i}}X_{b_{s,i}}. If C is not Kunz, these quadrics generate I(C′). If C is Kunz but not trigonal, three additional cubic generators involving the monomial X_{3γ/2} appear. In the trigonal, base‑point‑free case, a further family of cubic and quartic generators indexed by parameters k and r is required. The proof rests on a careful dimension count of the graded pieces I_n(C′), extending techniques of Stöhr for exceptional monomials.

The paper also discusses the relationship with Green’s conjecture. While K_{0,2}=0 (Noether’s theorem) holds for any integral non‑hyperelliptic curve, the equivalence K_{1,2}=0 ⇔ Cliff(C)>1 fails for non‑Gorenstein curves. The authors construct Kunz curves of arbitrarily high Clifford index for which K_{1,2}≠0, showing that the naive extension of Green’s conjecture does not survive singularities.

Finally, the authors pose several open problems: (1) whether the full statement of Theorem 1(II) holds without the unicuspidal monomial hypothesis; (2) a description of the moduli of nearly Gorenstein rational curves with prescribed semigroups; (3) a characterization of curves with Clifford index equal to one in the singular setting. These questions point toward a broader program of understanding canonical embeddings, syzygies, and Clifford theory for singular algebraic curves.