Effective results on projective normality of the first and second secant varieties

Technical Summary: Effective Results on Projective Normality of the First and Second Secant Varieties

Problem Statement
Let $X$ be a smooth projective complex variety of dimension $n$, embedded in $\mathbb{P}^r$ via the complete linear system of a very ample line bundle $L$. The $k$-th secant variety, $\Sigma_k(X, L)$, is the closure of the union of linear spans of $k+1$ points on $X$. A central problem in algebraic geometry is determining the geometric and cohomological properties of these secant varieties, specifically their singularities and the generation of their defining ideals.

Previous work by Chou–Song, Ullery, Ein–Niu–Park, and Choi–Lacini–Sheridan–Park established that for sufficiently positive line bundles, $\Sigma_k$ possesses mild singularities (normal Du Bois) and is projectively normal. However, these results were often non-effective or lacked explicit bounds on the positivity required. Furthermore, while it is known that the homogeneous ideal $I(\Sigma_k)$ is generated in degree $k+2$ under certain conditions, effective criteria for this generation, and specifically for generation by $3 \times 3$-minors of a catalecticant matrix (a conjecture by Eisenbud–Koh–Stillman and Sidman–Smith), remained to be fully quantified for higher dimensions.

This paper aims to establish effective bounds on the positivity of the embedding line bundle $L$ to ensure:

1. Projective normality and normal Du Bois singularities for the first ($\Sigma_1$) and second ($\Sigma_2$) secant varieties.
2. Generation of the defining ideal $I(\Sigma_1)$ by cubics.
3. Generation of $I(\Sigma_1)$ by $3 \times 3$-minors of a matrix of linear forms (the catalecticant matrix).

The authors assume $n \ge 2$, as sharp effective results for curves ($n=1$) were previously established by Ein–Niu–Park and Agostini–Park.

Methodology
The proofs rely on a cohomological approach developed in prior works by the authors and their collaborators. The strategy involves reducing the geometric properties of the secant varieties to cohomology vanishing theorems for vector bundles on Hilbert schemes of points and nested Hilbert schemes of points.

Key technical components include:

• Hilbert Schemes: The authors utilize the Hilbert scheme of $k$ points, $X[k]$, and the nested Hilbert scheme $X[k_1, \dots, k_\ell]$. They consider the tautological bundles $E_{k,L}$ associated with the line bundle $L$.
• Resolution of Singularities: The secant variety $\Sigma_k$ is realized as the image of a birational morphism $\alpha_k: B_k \to \Sigma_k$, where $B_k = \mathbb{P}(E_{k+1, L})$ is a projective bundle over $X[k+1]$. The singularities of $\Sigma_k$ are analyzed via the exceptional divisors of this resolution.
• Cohomology Vanishing: The core of the argument involves proving the vanishing of higher cohomology groups $H^i$ for specific vector bundles on $X[k]$ and nested schemes. These vanishing theorems are derived using the Kawamata–Viehweg vanishing theorem.
• Positivity Conditions: The line bundle $L$ is assumed to be of the form $L = \omega_X \otimes H^m \otimes B$, where $H$ is very ample, $B$ is nef, and $m$ is an integer. The authors determine the minimal integer $m$ required to ensure the necessary bundles are "nef and big" or satisfy specific ampleness conditions (e.g., $(k-1)$-very ample) to trigger vanishing.
• Singularities of Schemes: A crucial ingredient is the fact that the nested Hilbert schemes $X[1,2,3]$ and $X[1,2,3,4]$ possess Gorenstein canonical singularities, a result established in previous literature ([7], [8]). This property is essential for applying vanishing theorems and analyzing the Du Bois nature of the singularities.

Key Contributions and Results

1. Effective Bounds for Projective Normality and Singularities (Theorem 1.1)
The paper establishes that for $n \ge 2$, if $L = \omega_X \otimes H^m \otimes B$ with $m \ge 2n + 2$ (for $k=1$) or $m \ge 3n + 3$ (for $k=2$), then:

• $\Sigma_k$ has normal Du Bois singularities.
• $\Sigma_k$ is projectively normal.
• Higher cohomology groups $H^i(\Sigma_k, \mathcal{O}_{\Sigma_k}(\ell))$ vanish for $i > 0, \ell > 0$.
• $\Sigma_k$ is Cohen–Macaulay (resp. has rational singularities) if and only if $H^i(X, \mathcal{O}_X) = 0$ for $1 \le i \le n-1$ (resp. $1 \le i \le n$).

This result improves upon previous non-effective bounds and confirms that $m \ge 2n + 2$ suffices for the projective normality of $\Sigma_1$.

2. Generation by Cubics (Theorem 1.2)
The authors provide an effective condition for the homogeneous ideal $I(\Sigma_1)$ to be generated by cubics.

• Result: If $m \ge 3n + 2$, then $I(\Sigma_1(X, L))$ is generated by cubics.
• Method: This is proven by showing the vanishing of specific Koszul cohomology groups $K_{1,q}(\Sigma_1, \mathcal{O}_{\Sigma_1}(1))$ for $q \ge 3$, which reduces to cohomology vanishing on $X[2]$ involving exterior powers of the kernel bundle $M_{2,L}$.

3. Determinantal Presentation (Theorem 1.3)
The paper makes the conjecture of Agostini–Park (that $I(\Sigma_k)$ is generated by $(k+2) \times (k+2)$-minors of a catalecticant matrix) effective for $k=1$.

• Result: Let $L = L_1 \otimes L_2$ where $L_i = \omega_X \otimes H^{m_i} \otimes B_i$. If $m_1, m_2 \ge 3n + 2$, then $I(\Sigma_1(X, L))$ is generated by the $3 \times 3$-minors of the catalecticant matrix $Cat(L_1, L_2)$.
• Method: The proof demonstrates the surjectivity of a linear map from the tensor product of global sections to the degree 3 part of the ideal. This surjectivity is reduced to cohomology vanishing on $X[2]$ and $X[2,3]$ involving symmetric and exterior powers of kernel bundles.

Significance
The paper provides effective, quantitative thresholds for the positivity of line bundles that guarantee desirable geometric properties of secant varieties. By moving from qualitative "sufficiently positive" statements to explicit bounds involving the dimension $n$, the results make the theory applicable to concrete computations and specific varieties.

Specifically, the paper:

• Refines the understanding of the singularities of secant varieties, confirming they are Du Bois under explicit conditions.
• Resolves the effective version of the conjecture regarding the determinantal presentation of the first secant variety for $n \ge 2$, extending previous results known for curves or specific embeddings (like Veronese).
• Unifies the approach to projective normality, ideal generation, and singularity analysis through a robust cohomological framework on Hilbert schemes, leveraging the Gorenstein canonical nature of nested Hilbert schemes.

The authors note that their results are consistent with and generalize earlier findings for curves and specific cases, while providing the first effective bounds for higher-dimensional varieties in the context of the determinantal presentation conjecture.