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Asymptotic syzygies of secant varieties of curves. (English) Zbl 1494.14060

Let \(C\) be a smooth curve of genus \(g \geq 1\) over an algebraically closed field \(k\) of characteristic zero, embedded in \(\mathbb{P}H_0(C, L_d) = \mathbb{P}^{r_d}\) by a complete linear system of degree \(d\), and denote by \(\Sigma_k(C, L_d)\) the secant variety of \(k\)-planes. The main result of the paper consists in proving that minimal free resolution of this secant variety is asymptotically pure. As the author states in the introduction, an analogue of Erman’s theorem is presented, “that is, the Boij-Söderberg decomposition of the Betti table of the homogeneous coordinate ring of \(\Sigma_k(C, L_d)\) is increasingly dominated by a single pure diagram as \(d\) grows”. Finally, and as a corollary, it is also proved that the Betti numbers of the secant variety are normally distributed.

MSC:

14Q15 Computational aspects of higher-dimensional varieties
14Q20 Effectivity, complexity and computational aspects of algebraic geometry
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