[Paper Review] Cycle map on Hilbert schemes of nodal curves

Ziv Ran|ArXiv.org|Oct 2, 2004

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper establishes that the cycle map from the relative Hilbert scheme of m points on a family of nodal curves to the relative symmetric product is isomorphic to the blow-up of the discriminant locus—i.e., the locus of cycles with multiple points. The result is proven via local analysis of the map, showing it is a small resolution of singularities with fibers consisting of chains of rational curves. This provides a foundational geometric structure for studying Hilbert schemes in birational geometry and intersection theory.

ABSTRACT

We study the structure of the relative Hilbert scheme for a family of nodal (or smooth) curves via its natural cycle map to the relative symmetric product. We show that the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We discuss some applications and connections, notably with birational geometry and intersection theory on Hilbert schemes of smooth surfaces. Revised version corrects some minor errors.

Motivation & Objective

• To understand the geometric structure of the relative Hilbert scheme $X^{[m]}_B$ for families of nodal curves via its cycle map to the relative symmetric product.
• To show that the cycle map $\mathfrak{c}_m$ is the blow-up of the discriminant locus $D^m$ in $X^{(m)}_B$, where cycles have multiple points.
• To apply this result to study the canonical bundle, Euler characteristic, and Chern classes of tautological bundles on $X^{[m]}_B$.
• To connect the geometry of the Hilbert scheme to classical enumerative geometry and birational geometry, particularly through the lens of flips and flops.
• To provide a new derivation of Lehn’s formula for Chern classes of tautological bundles using the blow-up structure of the cycle map.

Proposed method

• The main method is a local study of the cycle map $\mathfrak{c}_m$ over the relative symmetric product $X^{(m)}_B$, analyzing its fibers and singularities.
• The proof uses the fact that $\mathfrak{c}_m$ is an isomorphism away from the discriminant locus $D^m$, and shows it is the blow-up of $D^m$ via ideal-theoretic and Proj constructions.
• The paper employs the flag Hilbert scheme $W^m(X/B)$ as a parameter space for filtered subschemes, with projections $p_i$ to the supports of the successive quotients.
• It uses the key identity $c(w^*\lambda_m(L)) = \prod_{j=1}^m (1 + L_j - \sum_{i=1}^{j-1} \Delta_{ij})$, where $\Delta_{ij}$ are the loci where the $i$-th and $j$-th points coincide.
• The derivation of the Chern class formula proceeds by expanding this product into monomials, grouping by connected components, and applying the pushforward $w_*$ to recover the tautological bundle classes.
• The method leverages Nakajima’s creation operators and the compatibility of the cycle map with the standard operations on cohomology, including the $\star$ multiplication on $X^{[m]}_B$.

Experimental results

Research questions

• RQ1 Is the cycle map $\mathfrak{c}_m: X^{[m]}_B \to X^{(m)}_B$ a blow-up of the discriminant locus $D^m$?
• RQ2 What is the structure of the fibers of $\mathfrak{c}_m$, particularly over multiple-point cycles?
• RQ3 How does the cycle map relate to birational geometry, especially in the context of flips and flops?
• RQ4 Can Lehn’s formula for Chern classes of tautological bundles be rederived from the blow-up structure of the cycle map?
• RQ5 What is the canonical bundle and Euler characteristic of the relative Hilbert scheme $X^{[m]}_B$?

Key findings

• The cycle map $\mathfrak{c}_m$ is the blow-up of the discriminant locus $D^m \subset X^{(m)}_B$, which parametrizes cycles with multiple points.
• The non-point fibers of $\mathfrak{c}_m$ are chains of rational curves, with at most $m-1$ components, confirming it is a small resolution of singularities.
• For $m=2$, the map $\mathfrak{c}_2$ is the Francia flip, and admits a natural 2:1 cover by the flop associated to a 3-fold ordinary double point.
• The canonical bundle of $X^{[m]}_B$ is computed and shown to behave like a flipping contraction, linking the geometry to birational geometry.
• A new formula for the Euler characteristic of $X^{[m]}_B$ is derived, based on the blow-up structure and the cycle map.
• A new derivation of Lehn’s formula for the Chern classes of $\lambda_m(L)$ is obtained by expanding the tautological class formula on the flag Hilbert scheme and pushing forward via $w_*$.