Powersum varieties, also called varieties of sums of powers, have provided examples of interesting relations between varieties since their first appearance in the 19th century. One of the most useful tools to study them is apolarity, a notion originally related to the action of differential operators on the polynomial ring. In this work, we make explicit how one can see apolarity in terms of the Cox ring of a variety. In this way, powersum varieties are a special case of varieties of apolar schemes; we explicitly describe examples of such varieties in the case of two toric surfaces, when the Cox ring is particularly well-behaved.
References
1.Â
Beauville, A., Complex Algebraic Surfaces, 2nd ed., London Mathematical Society Student Texts 34, Cambridge University Press, Cambridge, 1996. Translated from the 1978 French original by R. Barlow, with assistance from N.I. Shepherd-Barron and M. Reid.
2.Â
Catalisano, M., Geramita, A. and Gimigliano, A., Higher secant varieties of Segre-Veronese varieties, in Projective Varieties with Unexpected Properties, pp. 81–107, 2005.
3.Â
Cox, D., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17–50.
4.Â
Cox, D., Little, J. and Schenck, H., Toric Varieties, Graduate Studies in Mathematics 124, Am. Math. Soc., Providence, RI, 2011.
5.Â
Dolgachev, I., Dual homogeneous forms and varieties of power sums, Milan J. Math. 72 (2004), 163–187.
6.Â
Fisher, T., Pfaffian presentations of elliptic normal curves, Trans. Amer. Math. Soc. 362 (2010), 2525–2540.
7.Â
Fogarty, J., Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511–521.
8.Â
Friedman, R., Algebraic Surfaces and Holomorphic Vector Bundles, Universitext, Springer, Berlin, 1998.
9.Â
Fulton, W., Intersection Theory, 2nd ed., Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics 2, Springer, Berlin, 1998.
10.Â
Gała̧zca, M., Multigraded Apolarity (2016). arXiv:1601.06211
11.Â
Iarrobino, A. and Kanev, V., Power Sums, Gorenstein Algebras, and Determinantal Loci, Lecture Notes in Mathematics 1721, Springer, Berlin, 1999. Appendix C by Iarrobino and Steven L. Kleiman.
12.Â
Iliev, A. and Ranestad, K., surfaces of genus 8 and varieties of sums of powers of cubic fourfolds, Trans. Amer. Math. Soc. 353 (2001), 1455–1468.
13.Â
Kapranov, M., Chow quotients of Grassmannians. I, in I.M. Gel’fand Seminar, Adv. Soviet Math. 16, pp. 29–110, 1993.
14.Â
Landsberg, J., Tensors: Geometry and Applications, Graduate Studies in Mathematics 128, Am. Math. Soc., Providence, RI, 2012.
15.Â
Massarenti, A. and Mella, M., Birational aspects of the geometry of varieties of sums of powers, Adv. Math. 243 (2013), 187–202.
16.Â
Mukai, S., Fano 3-folds, in Complex Projective Geometry, Trieste, 1989/Bergen, 1989, London Math. Soc. Lecture Note Ser. 179, pp. 255–263, 1992.
17.Â
Oeding, L. and Ottaviani, G., Eigenvectors of tensors and algorithms for Waring decomposition, J. Symbolic Comput. 54 (2013), 9–35.
18.Â
Ranestad, K. and Schreyer, F., Varieties of sums of powers, J. Reine Angew. Math. 525 (2000), 147–181.
19.Â
Ranestad, K. and Schreyer, F., The variety of polar simplices, Doc. Math. 18 (2013), 469–505.
20.Â
Ranestad, K. and Voisin, C., Variety of power sums and divisors in the moduli space of cubic fourfolds, Doc. Math. 22 (2017), 455–504.
21.Â
Room, T., The Geometry of Determinantal Loci, Cambridge University Press, Cambridge, 1938.
22.Â
Sylvester, J., An essay on canonical forms, supplemented by a sketch of a memoir on elimination, transformation and canonical forms, in Collected Works, vol. I, pp. 203–216, Cambridge University Press, Cambridge, 1904.
23.Â
Sylvester, J., Sketch of a memoir on elimination, transformation, and canonical forms, in Collected Works, vol. I, pp. 184–197, Cambridge University Press, Cambridge, 1904.