Secant variety and syzygies of Hilbert scheme of two points

Technical Summary: The Secant Variety and Syzygies of the Hilbert Scheme of Two Points

Problem Statement
This paper investigates the geometric and algebraic properties of the Hilbert scheme of two points, denoted $X^{[2]}$, on a smooth projective variety $X$. Specifically, the authors analyze two primary aspects under the Grothendieck-Plücker embedding $\phi_{1,L}: X^{[2]} \hookrightarrow \mathbb{P}^N$ (induced by a line bundle $L$ on $X$):

1. Identifiability and Singularities: Determining the singular locus of the secant variety $\text{Sec}(X^{[2]})$. A central question is whether the non-identifiable locus (points lying on more than one secant or tangent line) coincides exactly with the variety $X^{[2]}$ itself, implying that $\text{Sec}(X^{[2]})$ is smooth outside $X^{[2]}$.
2. Syzygies: Establishing conditions under which the embedding of $X^{[2]}$ satisfies Green's condition $(N_p)$, which governs the linearity of the syzygies of the homogeneous coordinate ring.

The authors note that while secant varieties of highly symmetric spaces (like Grassmannians) or varieties with sufficiently positive embeddings have been studied, the specific behavior of the Hilbert scheme of points requires new analysis, particularly because the embedding of $X^{[2]}$ is not generally 2-very ample, complicating standard approaches.

Methodology
The authors employ a combination of deformation theory, Grassmannian geometry, and cohomological criteria for syzygies:

• Identifiability Analysis:

  • The authors utilize the classification of secant varieties of Grassmannians by Galgano and Staffolani [8], specifically the orbit stratification based on Hamming distance and tangent line ranks.
  • They establish technical lemmas (Lemmas 3.3–3.5) relating the global sections of line bundles to the inclusion of ideal sheaves of subschemes, provided the line bundle is sufficiently positive (specifically $d$-very ample).
  • They characterize lines in $X^{[d]}$ (Lemma 3.7), showing that a line connects two points $[Z_1], [Z_2]$ if and only if the subschemes differ at exactly one point.
  • The core of the proof involves a case-by-case analysis of intersections between secant lines, tangent lines, and pairs of tangent lines. They demonstrate that for $X^{[2]}$, any intersection of distinct secant lines or secant and tangent lines must occur within $X^{[2]}$ itself, provided $L$ is 4-very ample.

• Syzygy Analysis:

  • To prove condition $(N_p)$, the authors adapt the cohomological criterion of Ein and Lazarsfeld [5], which requires the vanishing of specific cohomology groups involving the syzygy bundle $M_L$.
  • They decompose the problem by analyzing the divisor $B \subset X^{[2]}$ parametrizing non-reduced subschemes, which is isomorphic to the projectivized cotangent bundle $\mathbb{P}(\Omega_X)$.
  • Using results from Park [20] on the syzygies of projective bundles, they verify the vanishing conditions on $B$ and extend these to the ambient space $X^{[2]}$ via restriction sequences and spectral sequences.

• Resolution of Singularities:

  • The authors construct a resolution of singularities using the secant bundle $\mathbb{P}(E_{L_1})$, where $E_{L_1}$ is a rank-2 vector bundle on $X^{[2]}$.
  • They analyze the map $u: \mathbb{P}(E_{L_1}) \to \text{Sec}(X^{[2]})$ by studying the fibers of the map induced by the Grothendieck-Plücker embedding, distinguishing between cases where the corresponding line lies in $X^{[2]}$ and where it does not.

Key Contributions and Results

1. Identifiability of $X^{[2]}$ (Theorem 1.1 & Theorem 3.1):
   The authors prove that if $L$ is a 4-very ample line bundle on $X$, the non-identifiable locus of $\text{Sec}(X^{[2]})$ is exactly $X^{[2]}$. This holds even though the embedding of $X^{[2]}$ is not generally 2-very ample. The proof relies on showing that secant and tangent lines to $X^{[2]}$ do not intersect outside of $X^{[2]}$.

2. Syzygies of $X^{[2]}$ (Theorem 1.2 & Theorem 4.3):
   The paper establishes that for a sufficiently positive line bundle $L$ (specifically of the form $K_X + dA + C$ with $d \gg 0$), the Grothendieck-Plücker embedding of $X^{[2]}$ satisfies Green's condition $(N_p)$ for any integer $p \geq 0$.

3. Singular Locus Characterization (Corollary 1.3):
   Combining the identifiability and syzygy results, the authors conclude that if $L$ is sufficiently positive, the singular locus of $\text{Sec}(X^{[2]})$ is exactly $X^{[2]}$.

4. Geometry of the Resolution (Proposition 1.4 & Proposition 5.4):
   For the case where $X$ is a surface, the authors describe the geometry of the resolution map $u: \mathbb{P}(E_{L_1}) \to \text{Sec}(X^{[2]})$. They show that the map factors through the blowup of the secant variety along $X^{[2]}$. The fibers of this resolution are:

   • Isomorphic to $\mathbb{P}^2$ if the point in the secant variety corresponds to a line contained in $X^{[2]}$ (i.e., points in the divisor $B$).
   • Reduced points otherwise.

Significance and Claims
The paper claims to shift the focus of secant variety analysis from highly symmetric varieties to the Hilbert scheme of points, utilizing the concept of identifiability as a primary tool for analyzing singularities. The authors emphasize that their results provide a complete description of the singular locus of $\text{Sec}(X^{[2]})$ under positivity conditions, a result that was previously unknown for this specific class of varieties.

The work highlights the essential role of syzygies in describing the resolution of singularities of secant varieties, noting that without conditions like $(N_p)$, the natural map from the secant bundle may not be an isomorphism to the blowup. The authors explicitly state that their results for the resolution geometry are currently limited to the surface case ($\dim X = 2$) due to technical difficulties in higher dimensions regarding the existence of higher-dimensional linear spaces in $X^{[2]}$.

The paper concludes by posing open questions regarding the identifiable locus of $X^{[3]}$, the existence of a specific bound for the positivity of $L$ required for the syzygy results, and the normality of $\text{Sec}(X^{[2]})$ when $X$ is a surface.