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Equations and syzygies of the first secant variety to a smooth curve

Author: Peter Vermeire
Journal: Proc. Amer. Math. Soc. 140 (2012), 2639-2646
MSC (2010): Primary 14N05, 14H99, 13D02
Posted: December 14, 2011
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that if is a linearly normal smooth curve embedded by a line bundle of degree at least , then the secant variety to the curve satisfies $N_{3,p}$ .

References

1. J. Alexander and A. Hirschowitz, Polynomial interpolation in several variables, J. Algebraic Geom. 4 (1995), no. 2, 201–222. MR 1311347 (96f:14065)
2. Elizabeth S. Allman and John A. Rhodes, Phylogenetic ideals and varieties for the general Markov model, Adv. in Appl. Math. 40 (2008), no. 2, 127–148. MR 2388607 (2008m:60145), http://dx.doi.org/10.1016/j.aam.2006.10.002
3. Aaron Bertram, Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space, J. Differential Geom. 35 (1992), no. 2, 429–469. MR 1158344 (93g:14037)
4. M. V. Catalisano, A. V. Geramita, and A. Gimigliano, On the ideals of secant varieties to certain rational varieties, J. Algebra 319 (2008), no. 5, 1913–1931. MR 2392585 (2009g:14068), http://dx.doi.org/10.1016/j.jalgebra.2007.01.045
5. David Cox and Jessica Sidman, Secant varieties of toric varieties, J. Pure Appl. Algebra 209 (2007), no. 3, 651–669. MR 2298847 (2008i:14077), http://dx.doi.org/10.1016/j.jpaa.2006.07.008
6. David Eisenbud, The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Springer-Verlag, New York, 2005. A second course in commutative algebra and algebraic geometry. MR 2103875 (2005h:13021)
7. David Eisenbud, Mark Green, Klaus Hulek, and Sorin Popescu, Restricting linear syzygies: algebra and geometry, Compos. Math. 141 (2005), no. 6, 1460–1478. MR 2188445 (2006m:14072), http://dx.doi.org/10.1112/S0010437X05001776
8. David Eisenbud, Jee Koh, and Michael Stillman, Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), no. 3, 513–539. MR 944326 (89g:14023), http://dx.doi.org/10.2307/2374621
9. T. Fisher, The higher secant varieties of an elliptic normal curve, preprint.
10. T. Fujita, Defining equations for certain types of polarized varieties, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 165–173. MR 0437533 (55 #10457)
11. Luis David Garcia, Michael Stillman, and Bernd Sturmfels, Algebraic geometry of Bayesian networks, J. Symbolic Comput. 39 (2005), no. 3-4, 331–355. MR 2168286 (2006g:68242), http://dx.doi.org/10.1016/j.jsc.2004.11.007
12. Adam Ginensky, A generalization of the Clifford index and determinantal equations for curves and their secant varieties, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–The University of Chicago. MR 2712616
13. Mark L. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), no. 1, 125–171. MR 739785 (85e:14022)
14. J. M. Landsberg and Jerzy Weyman, On secant varieties of compact Hermitian symmetric spaces, J. Pure Appl. Algebra 213 (2009), no. 11, 2075–2086. MR 2533306 (2010i:14095), http://dx.doi.org/10.1016/j.jpaa.2009.03.010
15. Robert Lazarsfeld, A sampling of vector bundle techniques in the study of linear series, Lectures on Riemann surfaces (Trieste, 1987) World Sci. Publ., Teaneck, NJ, 1989, pp. 500–559. MR 1082360 (92f:14006)
16. R. Lazarsfeld and A. Van de Ven, Topics in the geometry of projective space, DMV Seminar, vol. 4, Birkhäuser Verlag, Basel, 1984. Recent work of F. L. Zak; With an addendum by Zak. MR 808175 (87e:14045)
17. David Mumford, Varieties defined by quadratic equations, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) Edizioni Cremonese, Rome, 1970, pp. 29–100. MR 0282975 (44 #209)
18. M. S. Ravi, Determinantal equations for secant varieties of curves, Comm. Algebra 22 (1994), no. 8, 3103–3106. MR 1272376 (95c:14029), http://dx.doi.org/10.1080/00927879408825016
19. Jessica Sidman and Seth Sullivant, Prolongations and computational algebra, Canad. J. Math. 61 (2009), no. 4, 930–949. MR 2541390 (2010h:14084), http://dx.doi.org/10.4153/CJM-2009-047-5
20. Jessica Sidman and Peter Vermeire, Syzygies of the secant variety of a curve, Algebra Number Theory 3 (2009), no. 4, 445–465. MR 2525559 (2010j:13026), http://dx.doi.org/10.2140/ant.2009.3.445
21. Bernd Sturmfels and Seth Sullivant, Combinatorial secant varieties, Pure Appl. Math. Q. 2 (2006), no. 3, 867–891. MR 2252121 (2007h:14082)
22. Michael Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994), no. 2, 317–353. MR 1273268 (95e:14006), http://dx.doi.org/10.1007/BF01232244
23. Peter Vermeire, Some results on secant varieties leading to a geometric flip construction, Compositio Math. 125 (2001), no. 3, 263–282. MR 1818982 (2002c:14027), http://dx.doi.org/10.1023/A:1002663915504
24. Peter Vermeire, Secant varieties and birational geometry, Math. Z. 242 (2002), no. 1, 75–95. MR 1985451 (2004e:14025), http://dx.doi.org/10.1007/s002090100308
25. Peter Vermeire, Regularity and normality of the secant variety to a projective curve, J. Algebra 319 (2008), no. 3, 1264–1270. MR 2379097 (2009b:14099), http://dx.doi.org/10.1016/j.jalgebra.2007.10.032
26. P. Vermeire, Arithmetic properties of the secant variety to a projective variety, preprint.
27. Hans-Christian Graf v. Bothmer and Klaus Hulek, Geometric syzygies of elliptic normal curves and their secant varieties, Manuscripta Math. 113 (2004), no. 1, 35–68. MR 2135560 (2006b:14009), http://dx.doi.org/10.1007/s00229-003-0421-1

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