Hilbert polynomials of the first and second secant varieties

Technical Summary: Hilbert Polynomials of the First and Second Secant Varieties

Problem Statement
The paper addresses the computation of the cohomology groups of symmetric powers of the tautological bundle $E_{k+1,L}$ on the Hilbert scheme of points $X[k+1]$ for a smooth projective complex variety $X$ of dimension $n$. Specifically, the authors focus on the case where $k=1$ and $k=2$ (corresponding to Hilbert schemes of 2 and 3 points) and an arbitrary integer $\ell \geq 1$. The bundle $E_{k+1,L}$ is associated with a sufficiently positive line bundle $L$ on $X$.

While previous literature has extensively studied these cohomology groups for curves ($n=1$) or small values of $n$ and $\ell$, this work targets the generalization to arbitrary dimension $n$ and arbitrary symmetric power $\ell$, under the assumption that $L$ is sufficiently positive. The ultimate goal is to express the dimensions of these cohomology groups in terms of the Hilbert polynomials of the first and second secant varieties of the embedding defined by $L$.

Methodology
The authors employ a geometric approach linking the Hilbert scheme of points to secant varieties via projective bundles and birational morphisms.

1. Geometric Setup:

   • Let $B_k = \mathbb{P}(E_{k+1,L})$ be the projective bundle over $X[k+1]$. The complete linear system of the tautological line bundle $\mathcal{O}_{B_k}(1)$ induces a birational morphism $\alpha_{k+1}: B_{k+1} \to \Sigma_k$, where $\Sigma_k$ is the $k$-th secant variety of $X$ embedded in $\mathbb{P}H^0(X, L)$.
   • For $k=1, 2$, $\alpha_{k+1}$ serves as a resolution of singularities for $\Sigma_k$. The authors utilize the fact that for sufficiently positive $L$, the higher cohomology groups of the structure sheaf of the secant variety vanish ($H^i(\Sigma_k, \mathcal{O}_{\Sigma_k}(\ell)) = 0$ for $i>0, \ell>0$).

2. Leray Spectral Sequence and Direct Images:

   • The cohomology groups $H^i(X[k+1], S^\ell E_{k+1,L})$ are identified with $H^i(B_{k+1}, \mathcal{O}_{B_{k+1}}(\ell))$.
   • Using the Leray spectral sequence for the morphism $\alpha_{k+1}$, the problem reduces to computing the higher direct images $R^i \alpha_{k+1, *} \mathcal{O}_{B_{k+1}}$.
   • The authors explicitly compute these direct images by analyzing the exceptional divisors of the resolution. For $k=1$ (secant variety $\Sigma_1$), the exceptional divisor is the universal family $Z_2$ (the blow-up of $X \times X$ along the diagonal). For $k=2$ (secant variety $\Sigma_2$), the analysis involves a nested sequence of blow-ups and the universal family over $X[3]$.

3. Computation of Hilbert Polynomials:

   • Once the direct images are determined, the Euler characteristic $\chi(\mathcal{O}_{\Sigma_k}(\ell))$ (which equals the Hilbert polynomial $P_{\Sigma_k}(\ell)$ for $\ell \geq 1$) is computed via short exact sequences relating the structure sheaves of the secant varieties, their singular loci, and the projective bundles.
   • The authors utilize the K"unneth formula and properties of blow-ups (specifically relating to the exceptional divisors $F_1$ and $F_2$) to express these Euler characteristics in terms of the Euler characteristics of tensor products of symmetric powers of the cotangent bundle $\Omega_X$ and powers of $L$.
   • Finally, Lagrange interpolation is applied to the computed values at specific integer points to derive the general polynomial form for arbitrary $\ell$.

Key Results

• Theorem 1.1 (Cohomology Description):
  The paper provides explicit descriptions of the cohomology groups $H^i(X[2], S^\ell E_{2,L})$ and $H^i(X[3], S^\ell E_{3,L})$.

  • For $X[2]$, the cohomology is non-zero only for $0 \leq i \leq n$. The group $H^0$ corresponds to the sections of the line bundle on the first secant variety $\Sigma_1$, while higher groups are tensor products of $H^i(X, \mathcal{O}_X)$ and sections of $\mathcal{O}_X(\ell)$.
  • For $X[3]$, the cohomology extends up to $i=2n$. The groups are expressed as direct sums involving cohomology of $X$, $X[2]$, and sections of line bundles on $\Sigma_1$ and $\Sigma_2$.

• Theorem 1.2 (Hilbert Polynomials):
  The authors derive complete formulas for the Hilbert polynomials $P_{\Sigma_1}(\ell)$ and $P_{\Sigma_2}(\ell)$.

  • These polynomials are expressed in terms of:
    • $\Gamma = \sum (-1)^n h^p(X, \mathcal{O}_X)$ (related to the holomorphic Euler characteristic).
    • $l(a) = \chi(L^a)$.
    • $s_1(a, b) = \chi(S^a \Omega_X \otimes L^b)$.
    • $s_2(a, b, c) = \chi(S^a \Omega_X \otimes S^b \Omega_X \otimes L^c)$.
  • The formulas involve finite sums and products of these invariants, providing a closed-form expression for the Hilbert function of the secant varieties.

Significance and Claims
The paper claims to provide a higher-dimensional generalization of existing results found in the literature, specifically extending [10, Theorem 4.2] and [10, Theorem 4.4] which were restricted to the case where $X$ is a curve.

• Complement to Literature: The results complement existing studies that focused on small $n$ and $\ell$ by handling arbitrary $n$ and $\ell$ (for $k=1, 2$).
• Generalization of Degree Formulas: The authors note that while the degrees of $\Sigma_1$ and $\Sigma_2$ were previously known (citing [11] and [4]), their work offers a "far-reaching generalization" by computing the full Hilbert polynomials, not just the degrees.
• Methodological Contribution: The paper demonstrates a strategy for computing cohomology of symmetric powers of tautological bundles via secant varieties in higher dimensions, a task that requires substantially more involved arguments than the curve case (where Lagrange interpolation on a small range of values sufficed).

The authors maintain a modest tone regarding the scope, noting that the assumption of $L$ being "sufficiently positive" is necessary for the vanishing theorems used (Theorem 2.2) and that the results for $k=1, 2$ are the primary focus, with a brief discussion on extending the method to surfaces with arbitrary $k$ in Remark 3.2.