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Geometric syzygies of elliptic normal curves and their secant varieties. (English) Zbl 1053.14032

The paper under review deals with the following general question. To what extent the geometry of a given special subvariety \(X\subset\mathbb P^n\) determines the shape of the minimal free resolution of the ideal sheaf \(I_X\)? The starting point is a conjecture of M. Green [J. Differ. Geom. 19, 129–171 (1984; Zbl 0559.14008)] according to which, if \(C\subset\mathbb P^{g-1}\) is a canonical curve of genus \(g\geq 3\) with no geometric \(p\)-th linear syzygies, then \(C\) should not have linear syzygies at all. This conjecture has been solved by M. Teixidor [Duke Math. J. 111, 195–222 (2002; Zbl 1059.14039)] and by C. Voisin [J. Reine Angew. Math. 387, 111–121 (1988; Zbl 0652.14012)] in some special cases.
Inspired by Green’s conjecture the authors of this paper ask the following question. Do the geometric \(p\)-syzygies span the spaces of all \(p\)-th syzygies of a given variety \(X\subset\mathbb P^n\)? The answer to this question is positive in a number of special cases, e.g. for rational normal curves or for rational normal scrolls, but there are examples when the answer is definitely negative. The authors prove that the answer to this question is also positive for elliptic normal curves. Then they prove an analogous result for higher secant varieties of elliptic normal curves. Finally, they use these results to solve the question for bielliptic canonical curves of Clifford index \(2\).

MSC:

14H45 Special algebraic curves and curves of low genus
13D02 Syzygies, resolutions, complexes and commutative rings
14H52 Elliptic curves

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