[Paper Review] Generation and syzygies of the first secant variety

Peter Vermeire|ArXiv.org|Sep 25, 2008

Genetics and Plant Breeding50 references3 citations

TL;DR

This paper establishes effective positivity conditions under which the first secant variety of a smooth variety satisfies the $N_{3,p}$ property, using cohomological vanishing techniques on the Hilbert scheme. For smooth curves, it proves that embedding degree $d \geq 2g + 3 + p$ ensures $\Sigma_1$ satisfies $N_{3,p}$, providing the best known effective bound and confirming part of a conjecture on syzygies of secant varieties.

ABSTRACT

Under certain effective positivity conditions, we show that the secant variety to a smooth variety satisfies $N_{3,p}$. For smooth curves, we provide the best possible effective bound on the degree $d$ of the embedding, $d\geq 2g+3+p$.

Motivation & Objective

• To establish effective degree bounds ensuring the first secant variety $\Sigma_1$ of a smooth curve satisfies the $N_{3,p}$ property.
• To extend the $N_{3,p}$ condition to higher-dimensional smooth varieties under positivity conditions on the line bundle embedding.
• To provide a cohomological framework using the Hilbert scheme and vector bundles to analyze syzygies of secant varieties.
• To confirm and strengthen a conjecture on the syzygetic structure of secant varieties, particularly for curves.
• To bridge geometric information on secant varieties with Green-Lazarsfeld syzygy theory via vanishing theorems on $\operatorname{Hilb}^{2}X$.

Proposed method

• Use the Koszul-type criterion: $\Sigma_1$ satisfies $N_{3,p}$ if $H^1(\Sigma_1, \wedge^a M_L(b)) = 0$ for $2 \leq a \leq p+1$, $b \geq 2$, where $M_L$ is the pullback of $\Omega_{\mathbb{P}^n}(1)$.
• Reinterpret cohomology vanishings on $\Sigma_1$ via pullback to the blow-up $Z = \operatorname{Bl}_\Delta(X \times X)$, using the double cover $d: Z \to \operatorname{Hilb}^2 X$.
• Apply the spectral sequence from [18, 5.8] to relate $\operatorname{Tor}$-vanishing to syzygy generation, reducing the problem to vanishing on $Z$.
• Use the isomorphism $H^i(Z, \wedge^{a-1+i} d^* M_L \otimes L \boxtimes L \otimes \mathcal{O}(-E_\Delta)) \cong H^i(X, N^*_{X/\mathbb{P}^n} \otimes \wedge^{a-1+i} M_L \otimes L^2)$ for $i=1,2$.
• Prove vanishing of $H^i(X, N^*_{X/\mathbb{P}^n} \otimes \wedge^{a-1+i} M_L \otimes L^2)$ using positivity of adjoint line bundles and ampleness of $L$.
• Apply the fact that $\wedge^a M_L \otimes N^*(2)$ is a summand of a tensor power of $M_L$, allowing use of Kodaira-type vanishing under big and nef conditions.

Experimental results

Research questions

• RQ1 What is the optimal effective degree bound on the line bundle embedding of a smooth curve such that its first secant variety satisfies $N_{3,p}$?
• RQ2 Under what positivity conditions on the line bundle $L$ does the first secant variety $\Sigma_1$ of a smooth variety satisfy $N_{3,p}$?
• RQ3 Can cohomological vanishings on the Hilbert scheme $\operatorname{Hilb}^2 X$ be used to deduce syzygetic properties of secant varieties?
• RQ4 How do the syzygies of $\Sigma_1$ relate to the syzygies of the original variety $X$ under adjoint linear systems?
• RQ5 Is the conjectured $N_{3,p}$ property for $\Sigma_1$ optimal, and can it be extended to higher-dimensional varieties?

Key findings

• For a smooth curve $C$ of genus $g$ embedded by a line bundle of degree $d \geq 2g + 3 + p$, the first secant variety $\Sigma_1$ satisfies $N_{3,p}$.
• The bound $d \geq 2g + 3 + p$ is optimal and effective, improving upon previous results that required $d \geq 4g + 2k + 3$ for similar conclusions.
• For any smooth variety $X$ embedded by a sufficiently positive line bundle $L = M^k$ with $k \gg 0$, the first secant variety $\Sigma_1$ satisfies $N_{3,p}$.
• When $X$ is not $\mathbb{P}^d$, and $L = K_X \otimes M^{(p+2)d+1}$ with $K_X \otimes M$ ample, $\Sigma_1$ satisfies $N_{3,p}$, providing an effective construction via adjoint systems.
• The cohomological criterion $H^1(\Sigma_1, \wedge^a M_L(b)) = 0$ for $2 \leq a \leq p+1$, $b \geq 2$ is sufficient for $\Sigma_1$ to satisfy $N_{3,p}$, and this is reduced to vanishings on $X$ via the Hilbert scheme.
• The paper confirms Conjecture 1.1(3) for $k=1$, showing $\Sigma_1$ satisfies $N_{3,p}$ under the stated degree bounds for curves.