On infinitesimal deformations of singular varieties III

We study the affine cone over a reducible nodal curve $X$ obtained by gluing three projective lines along three pairs of points to form a connected curve of arithmetic genus (1). We endow (X) with a line bundle (L) of multidegree ((4,3,3)), and we show that (L) is very ample, giving an embedding into ( \mathbb{P}^9). We then analyze in detail the affine cone ( C(X) ) and determine its singular locus, which consists of three singular lines meeting at the vertex.

💡 Research Summary

The paper investigates the affine cone over a reducible nodal curve of arithmetic genus one, providing a complete description of its deformation theory. The curve X is constructed by taking three copies of ℙ¹ (C₁, C₂, C₃) and gluing them pairwise at three distinct points: a₁∈C₁ with c₁∈C₃, a₂∈C₁ with b₂∈C₂, and b₁∈C₂ with b₃∈C₂. This yields a connected curve with three nodes, whose dual graph contains a single loop, giving pₐ(X)=1.

A line bundle L on X with multidegree (4, 3, 3) is introduced. On each component Cᵢ, L restricts to O_{ℙ¹}(dᵢ) with dᵢ≥3, which is very ample and separates 2‑jets. By analyzing sections via the normalization map ν: \tilde X→X, the authors write a global section of L as a triple (s₁,s₂,s₃)∈H⁰(C₁,O(d₁))⊕H⁰(C₂,O(d₂))⊕H⁰(C₃,O(d₃)) subject to three linear matching conditions at the nodes. Solving these linear equations shows that h⁰(X,L)=deg L=10, so the complete linear series |L| embeds X as a projectively normal curve in ℙ⁹.

The affine cone C(X)=Spec ⊕_{m≥0} H⁰(X,L^{⊗m}) is shown to be normal. Its singular locus consists of the vertex together with three one‑dimensional components, each lying over one of the nodes; the three lines meet at the vertex, forming a non‑isolated surface singularity.

The core of the paper is the computation of the graded pieces of the deformation space T¹(C(X)) under the natural Gₘ‑action, following Pinkham’s theory. For each integer m, a line bundle Fₘ on X is defined:

• m>0: Fₘ = T_X ⊗ L^{⊗m}
• m=0: F₀ = T_X
• m<0: Fₘ = L^{⊗m}

Pinkham’s result gives T¹(C(X))(m) ≅ H¹(X,Fₘ) for m≥0 and T¹(C(X))(m) ≅ H⁰(X,Fₘ) for m<0. The authors compute these cohomology groups explicitly. For a generic choice of gluing data, H⁰(X,T_X)=H¹(X,T_X)=0, which forces T¹(C(X))(0)=0; thus there are no degree‑zero equivariant deformations. Moreover, for all m>0, H¹(X,T_X⊗L^{⊗m}) vanishes, so positive‑weight deformations are absent.

For negative weights, H⁰(X,L^{⊗m}) is non‑zero. In particular, dim T¹(C(X))(-1)=10, and each negative weight contributes a vector space of dimension –10 m. The authors construct an explicit one‑parameter deformation 𝒞→Δ (Δ a small disk) whose Kodaira–Spencer class lies in T¹(C(X))(-1). For t≠0 the fiber 𝒞_t is smooth; the deformation simultaneously smooths the three singular lines and the vertex. Hence the cone admits a complete smoothing governed solely by a degree‑negative deformation.

The paper situates this example at the intersection of several moduli problems: (i) the boundary point of the moduli space M₁ of stable genus‑one curves, (ii) the moduli of polarized reducible curves (X,L), (iii) the deformation space of Gₘ‑equivariant surface singularities, and (iv) smoothings of non‑isolated singularities arising from cones. It demonstrates that the geometry of a reducible nodal curve can fully control the deformation theory of its cone, providing a rare, fully computable instance of a non‑isolated surface singularity whose deformations are dictated by the underlying curve’s Picard group.

In summary, the authors prove that the affine cone over the chosen reducible genus‑one curve is very rigid in degree zero, has no positive‑weight deformations, but possesses a rich family of negative‑weight deformations that yield a complete smoothing. This explicit calculation enriches our understanding of graded deformation theory, offers a concrete model for studying non‑isolated singularities, and may have applications in areas such as birational geometry, compactifications of moduli spaces, and even theoretical physics contexts where cone singularities appear.