Secant Varieties of Segre-Veronese Varieties
Secant varieties of Segre and Veronese varieties (and more generally Segre-Veronese varieties, which are embeddings of a product of projective spaces via the complete linear system of an ample line bundle) are very classical objects that go back to the Italian school of mathematics in the 19-th century. Despite their apparent simplicity, little is known about their equations, and even less about the resolutions of their coordinate rings. The main goal of this thesis is to introduce a new method for analyzing the equations and coordinate rings of the secant varieties to Segre-Veronese varieties, and to work out the details of this method in the first case of interest: the variety of secant lines to a Segre-Veronese variety.
There is an extensive literature explaining the advantages of analyzing the equations of the secant varieties of a subvariety X of the projective space P^N as G-modules, when X is endowed with a G-action that extends to P^N. For X a Segre-Veronese variety, the corresponding G is a general linear (GL) group, or a product of such. Looking inside the highest weight spaces of carefully chosen GL-representations, we identify a set of ``generic equations'' for the secant varieties of Segre-Veronese varieties. The collections of ``generic equations'' form naturally modules over (products of) symmetric groups and moreover, they yield by the process of specialization all the (nongeneric) equations of the secant varieties of Segre-Veronese varieties.
Once we reduce our problem to the analysis of ``generic equations'', the representation theory of symmetric groups comes into play, and with it the combinatorics of tableaux. In the case of the first secant variety of a Segre-Veronese variety, we are naturally led to consider 1-dimensional simplicial complexes, i.e. graphs, attached to the relevant tableaux. We believe that simplicial complexes should play an important role in the combinatorics that emerges in the case of higher secant varieties.
The results of this thesis go in two directions. For both of them, the reduction to the ``generic'' situation is used in an essential way. One direction is showing that if we put together the 3x3 minors of certain generic matrices (called flattenings), we obtain a generating set for the ideal of the secant line variety of a Segre-Veronese variety. In particular, this recovers a conjecture of Garcia, Stillman and Sturmfels, corresponding to the case of a Segre variety. We also give a representation theoretic description of the homogeneous coordinate ring of the secant line variety of a Segre-Veronese variety. In the cases when this secant variety fills the ambient space, we obtain formulas for decomposing certain plethystic compositions.
A different direction is, for the Veronese variety, to show that for k small, the ideal of kxk minors of the various flattenings (which in this case are also known as catalecticant matrices) are essentially independent of which flattening we choose. In particular this proves a conjecture of Geramita, stating that the ideals of 3x3 minors of the ``middle'' catalecticant matrices are the same, and moreover that the ideal of the first secant variety of a Veronese variety is generated by the 3x3 minors of any such catalecticant.