Hilbert Scheme of Locally Cohen-Macaulay Curves

Introduction to Hilbert Schemes

Hilbert schemes are fundamental objects in algebraic geometry that parametrize subschemes of a projective space with a given Hilbert polynomial. Introduced by Alexander Grothendieck in his 1961 seminar at Harvard, Hilbert schemes have become essential tools in studying moduli problems and geometric constructions.

The Hilbert scheme H represents a moduli space that parametrizes all closed subschemes of a projective space P^n with a fixed Hilbert polynomial P. For each subscheme Z P^n with Hilbert polynomial P, H contains a point corresponding to Z. This construction provides a powerful way to organize the vast landscape of geometric objects sharing certain numerical invariants.

Locally Cohen-Macaulay Curves

A curve X is called locally Cohen-Macaulay if for every point p X, the local ring O_{X,p} is a Cohen-Macaulay ring. This condition means that the depth of the local ring equals its dimension. For curves, being Cohen-Macaulay is equivalent to having no embedded points.

Locally Cohen-Macaulay curves are particularly nice to study because they avoid pathologies like embedded components, which can cause technical difficulties. While every smooth curve is Cohen-Macaulay, many singular curves also satisfy this property. For instance, curves with only ordinary double points or nodal singularities are Cohen-Macaulay.

One important characterization: a curve X in P^n is locally Cohen-Macaulay if and only if it has no embedded points, which is equivalent to the Hartshorne-Rao module H^1_*(I_X) being zero. This characterization will be crucial when discussing Hilbert schemes of such curves.

Hilbert Scheme of Locally Cohen-Macaulay Curves

The Hilbert scheme of locally Cohen-Macaulay curves in P^n with given Hilbert polynomial P is a subscheme of the full Hilbert scheme H_{P,P^n}. We denote it by H_{LCM,P^n}^P. This subscheme parametrizes only those curves that are locally Cohen-Macaulay.

For curves, the Hilbert polynomial is of the form P(t) = dt + 1 - g, where d is the degree of the curve and g is its arithmetic genus. Therefore, the Hilbert scheme of locally Cohen-Macaulay curves of degree d and genus g in P^n is often denoted by H_{d,g,n}(LCM).

The existence of irreducible components in these Hilbert schemes is a classical problem in algebraic geometry. A component is called "nice" if it contains the smooth curves (when they exist). Understanding whether a given Hilbert polynomial corresponds to a component that contains smooth curves, curves with certain singularities, or more pathological examples is central to the study of these schemes.

Hartshorne's conjectures, developed in the 1970s, provided important insights into these Hilbert schemes. He conjectured that for many values of d, g, and n, the Hilbert scheme would be irreducible with a smooth curve as a general element. These conjectures have driven much research in the field.

Important Properties

Irreducibility Results

A central question in the study of Hilbert schemes is understanding their irreducible components. Many results in this area address when the Hilbert scheme of locally Cohen-Macaulay curves is irreducible:

For curves of low degree or high genus, the situation can be more complex. Elliptic normal curves (smooth curves of genus 1 in P^n with degree n+1) provide interesting examples. While smooth elliptic normal curves always exist, the Hilbert schemes containing them may have other components that consist of singular or degenerate curves.

Connectedness Results

While irreducibility may not always hold, many Hilbert schemes are connected. This means there is a continuous family of curves connecting any two curves in the scheme, even if some intermediate curves might have worse singularities or embedded components.

Tangent Space Interpretation

The tangent space at a point [X] in the Hilbert scheme H_{LCM,P^n}^P can be identified with Hom(I_X, O_X), where I_X is the ideal sheaf of X. For locally Cohen-Macaulay curves, this has a particularly nice interpretation as the space of first-order deformations of X in P^n.

This identification helps study the smoothness of the Hilbert scheme at points corresponding to certain curves. For example, if X is a locally complete intersection curve, then H_{LCM,P^n}^P is smooth at [X].

Dimensional Analysis

The expected dimension of H_{d,g,n}(LCM) is given by the Riemann-Roch formula for curves in P^n:

h^0(N_{X/P^n}) h^1(N_{X/P^n}) = (n+1)d + (n-3)(1-g)

where N_{X/P^n} is the normal bundle of X in P^n. When the actual dimension equals this expected dimension, we say the component is nonspecial. Components with larger dimension are called special or non-reduced.

Applications and Current Research

The study of Hilbert schemes of locally Cohen-Macaulay curves has numerous applications in algebraic geometry and related fields:

Classification of Algebraic Curves

Hilbert schemes provide a framework for classifying curves based on their degree, genus, and embedding dimension. Understanding the components of these schemes reveals the zoo of possible curves and how they relate to each other through families.

Fano Schemes

The Fano scheme of a projective variety X is the Hilbert scheme parameterizing lines contained in X. Knowledge of Hilbert schemes of curves is essential for understanding Fano schemes, which in turn provide information about the geometry of X.

Moduli of Vector Bundles

There is a deep connection between Hilbert schemes of curves and moduli spaces of vector bundles, particularly stable rank-2 vector bundles on P^2. This connection arises from the Serre correspondence between vector bundles and curves in projective spaces.

Recent Advances

Recent research has focused on extending classical results to more general settings, such as curves with richer singularities or curves embedded in other ambient spaces besides projective space. Techniques from derived algebraic geometry and moduli theory have provided new tools for studying these Hilbert schemes.

Current open problems include determining the number of irreducible components of Hilbert schemes for given values of d, g, and n, understanding their smoothness properties, and connecting these schemes to other geometric constructions like nested Hilbert schemes (parametrizing pairs of nested curves).

References

1. Hartshorne, R. "Connectedness of the Hilbert Scheme." Publications Mathmatiques de l'IHS, 1966.
2. Hartshorne, R. "Algebraic Geometry." Springer, 1977.
3. Gruson, L., and Peskine, C. "Genre des courbes de l'espace projectif." In: Algebraic geometry, Troms 1977.
4. Sernesi, E. "Deformations of Algebraic Schemes." Springer, 2006.
5. Martin-Deschamps, M. and Perrin, D. "Le schma de Hilbert des courbes gauches localement Cohen-Macaulay n'est pas (ncessairement) rduit." Ann. Sci. cole Norm. Sup., 2001.