Hilbert Scheme of Conics

• Hilbert scheme of conics is a moduli space that parameterizes closed subschemes with Hilbert polynomial 2m+1, capturing both smooth and singular conics.
• It is constructed via Grassmannian embeddings and Plücker equations, providing an explicit and computationally tractable framework in projective spaces.
• The scheme reveals deep insights into deformation theory, group actions, and derived category methods with applications to Fano and del Pezzo varieties.

A Hilbert scheme of conics is a fine moduli space parameterizing closed subschemes of a given ambient variety—typically a projective variety $X$—whose Hilbert polynomial equals $2m+1$. This scheme, denoted $\Hilb^{2m+1}(X)$, systematically organizes both smooth and singular conics including non-reduced structures and broken conic configurations. In higher-dimensional settings, such as Fano varieties, adjoint varieties, and complete intersections, the Hilbert scheme of conics reveals subtle aspects of deformation theory, intersection theory, and group actions, with rich connections to automorphism groups and derived category methods.

1. General Definition and Embedding

For $X$ a projective variety over $\mathbb{C}$, the Hilbert scheme of conics is the closed subscheme of $\Hilb(X)$ representing families of subschemes $C\subset X$ with the Hilbert polynomial

$\chi(\mathcal{O}_C(m)) = 2m + 1.$

This includes both smooth conics (irreducible degree-2 curves) and singular geometries (unions of lines, double lines, etc.). The space $\Hilb^{2m+1}(X)$ can often be realized as a subscheme of a Grassmannian via Gotzmann’s regularity. For $X = \mathbb{P}^n$, every conic ideal is $2$-regular, and thus

$\Hilb^{2m+1}(\mathbb{P}^n) \hookrightarrow \mathrm{Gr}_5(N(2)),$

where $N(2)$ is the dimension of $H^0(\mathbb{P}^n, \mathcal{O}(2))$ and $5$ arises from the value $p(2) = 5$ for $p(m) = 2m+1$ (Brachat et al., 2011). The Plücker embedding defines explicit equations of degree at most $d+2=4$ in the Plücker coordinates, providing an explicit and computationally tractable description.

2. Structure and Properties: Smoothness, Dimension, and Bundle Descriptions

In classical projective space, $\Hilb^{2m+1}(\mathbb{P}^n)$ is a smooth, irreducible projective variety of dimension $3n-1$ and is analytically isomorphic to a projective bundle over the Grassmannian of planes: $\Hilb^{2m+1}(\mathbb{P}^n) \cong \mathbb{P}(\mathrm{Sym}^2 U^*)$ where $U$ is the tautological rank-3 subbundle on $\mathrm{Gr}(3, n+1)$ (Brachat et al., 2011).

For low-degree complete intersections $X\subset\mathbb{P}^n$ of multi-degree $(d_1,\dots,d_c)$ and codimension $c$, Pan’s construction shows that the space of conics through two general points $p$, $q$ is itself a smooth complete intersection of type $(1,1,2,\ldots, d_i)$, with explicit equations defined in ambient projective bundles. The boundary divisor, corresponding to nodal or reducible conics, is a simple normal crossing divisor and itself a complete intersection (Pan, 2013).

3. Quintic Del Pezzo Varieties: Fixed Loci and Deformation–Obstruction Theory

For the quintic del Pezzo 4-fold $X$, realized as a smooth linear section of $\mathrm{Gr}(2,5)$ by cutting with hyperplanes $H_1, H_{-1}$ in the Plücker embedding, the Hilbert scheme $H_2(X) := \Hilb^{2m+1}(X)$ is constructed and shown to be smooth of dimension $7$ (Chung et al., 7 Dec 2025). The proof utilizes a torus action on $X$, analyzing the fixed loci consisting of isolated points, fixed lines (free and non-free, classified by normal bundles), and fixed planes (Schubert types). Every torus-fixed conic is either a smooth, reducible, or non-reduced configuration. The deformation-obstruction theory is explicit:

• For smooth conics $C$, $$N_{C/X} \cong \mathcal{O}_{\mathbb{P}^1}(2) \oplus \mathcal{O}_{\mathbb{P}^1}(1)^{\oplus 2}$$, yielding $h^1=0$, $h^0=7$.
• For broken or double-line conics, Mayer-Vietoris and Čech cohomology show vanishing of obstructions.

Global smoothness follows by the Białynicki–Birula principle, and $H_2(X)$ is rational. The approach generalizes to all quintic del Pezzo varieties with torus actions presenting suitable isolated loci (Chung et al., 7 Dec 2025).

4. Group Actions, Faithfulness, and Derived Category Correspondences

For smooth Fano threefolds, the Hilbert scheme of conics $S(X) = \Hilb^{2m+1}(X)$ can be analyzed via automorphism group actions. The $\Aut(X)$-action on $S(X)$ is faithful except in explicit double-cover cases; for genus $g=10$ and Picard rank one, $S(X)$ is identified as an abelian surface $\Sigma(Y)$, where $Y$ is the unique index-2 Fano threefold of degree 4. The derived category perspective, using semiorthogonal decompositions, places conic ideal sheaves into a Calabi-Yau subcategory $\mathcal{A}_X$, with equivalences to moduli of line bundles on genus-2 curves (for $g=10$),

$S(X) \cong \operatorname{Pic}^0(Z),$

giving a geometric and categorical moduli interpretation (Kuznetsov et al., 2016).

5. Spherical Varieties and Contact Geometry for Conics in Adjoint Varieties

For adjoint varieties $Z_g\subset\mathbb{P}(\mathfrak{g})$ not of types $A$ or $C$, the normalization $\widetilde{H}$ of the component parametrizing all smooth conics is a spherical $G$-variety. The locus of twistor conics forms a dense, open orbit $G/K$ (with $K$ associated to an involution $\sigma:G\to G$). The colored fan of $\widetilde{H}$ is explicitly computed using the Luna–Vust theory, with colors indexed by half restricted roots and valuation cones expressed via fundamental weights. The conjugacy classes of conics correspond to $G$-orbits stratified as twistor, contact, reducible, and double-line conics, with dimensions and numbers depending on the Lie type. Smoothness is proved via both deformation-theoretic and spherical-embedding criteria (Kwon, 2023). In contrast, the normalization of the Chow scheme exhibits different singularity and stratification properties.

6. Explicit Equations: Grassmannian, Quiver, and Plücker Perspectives

The Hilbert scheme of conics in $\mathbb{P}^n$ admits several explicit presentations:

• As a subscheme of $\mathrm{Gr}_5(N(2))$ cut out by cubic equations in Plücker coordinates, via the extensor method and Gotzmann regularity (Brachat et al., 2011).
• As a GIT quotient of a quiver of linear maps, embedded into a Grassmannian and cut out by explicit linear and rank-one quadratic equations in Plücker coordinates, resembling Stiefel-type conditions. These equations arise from symmetric group actions on variables and monomials (Evain et al., 2016).
• Bundle presentations, where the Hilbert scheme is a projective bundle over a Grassmannian parameterizing the underlying planes of conics.

7. Generalizations, Conjectures, and Enumerative Outcomes

Methods developed for quintic del Pezzo varieties suggest broader phenomena. If a smooth Fano variety of index $\ge 2$ admits a torus action with isolated fixed loci of suitable type, then the Hilbert scheme of conics is expected to be smooth of the predicted dimension (Chung et al., 7 Dec 2025). Pan’s analysis on complete intersections yields enumerative formulas: e.g., for a cubic threefold in $\mathbb{P}^4$, exactly six conics pass through two general points, and for a 3-fold intersection of two quadrics in $\mathbb{P}^5$, precisely two conics pass through two general points (Pan, 2013). These explicit counts are derived from the complete intersection structure and Grothendieck–Riemann–Roch computations.

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In sum, the Hilbert scheme of conics serves as a robust moduli space encoding complex algebraic, geometric, and representation-theoretic data, with variant structure and properties dependent on the ambient variety. Its paper encompasses deformation theory, torus and group actions, combinatorial models, and categorial moduli interpretations, unified through explicit geometric and algebraic equations.