We show that the secant variety of a linearly normal smooth curve of degree at least 2g+3 is arithmetically Cohen-Macaulay, and we use this information to study the graded Betti numbers of the secant variety.Comment: 24 pages; minor revision and reorganizatio

We describe an elementary counterexample to a conjecture of Fulton on the set of effective divisors on the space M 0 n of marked rational curves.  2002 Elsevier Science (USA)

For a smooth curve of genus g embedded by a line bundle of degree at least 2g + 3 we show that the ideal sheaf of the secant variety is 5-regular. This bound is sharp with respect to both the degree of the embedding and the bound on the regularity. Further, we show that the secant variety is projectively normal for the generic embedding of degree at least 2g + 3. The resultWe work over an algebraically closed field of characteristic 0. Recall the classical theorem of Castelnuovo:Theorem 1. Let C ⊂ P n be a linearly normal embedding of a smooth curve of genus g by a line bundle L with c 1 (L) 2g + 1. Then I C is 3-regular and (equivalently) C ⊂ P n is projectively normal.The following extension was proved for a = 2 by J. Rathmann [10] and was proved in general by the author [14, 4.2] (see also [2]).

We show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations. These were first constructed as flips in the case of curves by M. Thaddeus via Geometric Invariant Theory, and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper. We expose the finer structure of a second flip; again for varieties of arbitrary dimension. We also prove a result on the cubic generation of the secant variety and give some conjectures on the behavior of equations defining the higher secant varieties.

Abstract. In the 1980's, work of Green and Lazarsfeld [10,11] helped to uncover the beautiful interplay between the geometry of the embedding of a curve and the syzygies of its defining equations. Similar results hold for the first secant variety of a curve, and there is a natural conjectural picture extending to higher secant varieties as well. We present an introduction to the algebra and geometry used in [20] to study syzygies of secant varieties of curves with an emphasis on examples of explicit computations and elementary cases that illustrate the geometric principles at work.