Characterizations of Veronese and Segre varieties

We survey the known and recent characterizations of Segre varieties and Veronesea varieties

Secant dimensions of low-dimensional homogeneous varieties

We completely describe the higher secant dimensions of all connected homogeneous projective varieties of dimension at most 3, in all possible equivariant embeddings. In particular, we calculate these dimensions for all Segre-Veronese embeddings of P^1 * P^1, P^1 * P^1 * P^1, and P^2 * P^1, as well as for the variety F of incident point-line pairs in P^2. For P^2 * P^1 and F the results are new, while the proofs for the other two varieties are more compact than existing proofs. Our main tool is the second author's tropical approach to secant dimensions.Comment: 25 pages, many picture

The Rough Veronese variety

We study signature tensors of paths from a geometric viewpoint. The signatures of a given class of paths parametrize an algebraic variety inside the space of tensors, and these signature varieties provide both new tools to investigate paths and new challenging questions about their behavior. This paper focuses on signatures of rough paths. Their signature variety shows surprising analogies with the Veronese variety, and our aim is to prove that this so-called Rough Veronese is toric. The same holds for the universal variety. Answering a question of Amendola, Friz and Sturmfels, we show that the ideal of the universal variety does not need to be generated by quadrics

New strings for old Veneziano amplitudes II. Group-theoretic treatment

In this part of our four parts work (e.g see Part I, hep-th/0410242) we use the theory of polynomial invariants of finite pseudo-reflection groups in order to reconstruct both the Veneziano and Veneziano-like (tachyon-free) amplitudes and the generating function reproducing these amplitudes. We demonstrate that such generating function can be recovered with help of the finite dimensional exactly solvable N=2 supersymmetric quantum mechanical model known earlier from works by Witten, Stone and others. Using the Lefschetz isomorphisms theorem we replace traditional supersymmetric calculations by the group-theoretic thus solving the Veneziano model exactly using standard methods of representation theory. Mathematical correctness of our arguments relies on important theorems by Shepard and Todd, Serre and Solomon proven respectively in early fifties and sixties and documented in the monograph by Bourbaki. Based on these theorems we explain why the developed formalism leaves all known results of conformal field theories unchanged. We also explain why these theorems impose stringent requirements connecting analytical properties of scattering amplitudes with symmetries of space-time in which such amplitudes act.Comment: 57 pages J.Geom.Phys.(in press, available on line