Last syzygies of 1-generic spaces

Consider a determinantal variety X of expected codimension definend by the maximal minors of a matrix M of linear forms. Eisenbud and Popescu have conjectured that 1-generic matrices M are characterised by the property that the syzygy ideals I(s) of all last syzygies s of X coincide with I_X. In this note we prove a geometric version of this characterization, i.e. that M is 1-generic if and only if the syzygy varieties Syz(s)=V(I(s)) of all last syzyzgies have the same support as X.Comment: AMS Latex, 11 Page

Generic Syzygy Schemes

For a finite dimensional vector space G we define the k-th generic syzygy scheme Gensyz_k(G) by explicit equations. We show that the syzygy scheme Syz(f) of any syzygy in the linear strand of a projective variety X which is cut out by quadrics is a cone over a linear section of a corresponding generic syzygy scheme. We also give a geometric description of Gensyz_k(G) for k=0,1,2. In particular Gensyz_2(G) is the union of a Pl"ucker embedded Grassmannian and a linear space. From this we deduce that every smooth, non-degenerate projective curve C which is cut out by quadrics and has a p-th linear syzygy of rank p+3 admits a rank 2 vector bundle E with det E = O_C(1) and h^0(E) at least p+4.Comment: 12 Pages. This paper is a completely rewritten version of the first part of math.AG/0108078. It also contains several new result

Green-Lazarsfeld's Conjecture for Generic Curves of Large Gonality

We use Green's canonical syzygy conjecture for generic curves to prove that the Green-Lazarsfeld gonality conjecture holds for generic curves of genus g, and gonality d, if $g/3<d<[g/2]+2$.Comment: 5 page

Path planning and optimization in the traveling salesman problem: Nearest neighbor vs. region-based strategies

According to the number of targets, route planning can be a very complex task. Human navigators, however, usually solve route planning tasks fastly and efficiently. Here two experiments are presented that studied human route planning performance, route planning strategies employed, and cognitive processes involved. For this, 25 places were arranged on a regular grid in a large room. Each place was marked by a unique symbol. Subjects were repeatedly asked to solve traveling salesman problems (TSP), i.e. to find the shortest closed loop connecting a given start place with a number of target places. For this, subjects were given a so-called \u27shopping list\u27 depicting the symbols of the start place and the target places. While the TSP is computationally hard, sufficient solutions can be found by simple strategies such as the nearest neighbor strategy. In Experiment 1, it was tested whether humans deployed the nearest neighbor strategy (NNS) when solving the TSP. Results showed that subjects outperformed the NNS in cases in which the NNS did not predict the optimal solution, suggesting that the NNS is not sufficient to explain human route planning behavior. As a second possible strategy a region-based approach was tested in Experiment 2. When optimal routes required more region transitions than other, sub-optimal routes, subjects preferred these sub-optimal routes. This result suggests that subjects first planned a coarse route on the region level and then refined the route during navigation. Such a hierarchical planning stragey would allow to reduce computational effort during route planning. In a control condition, the target places were directly marked in the environment rather than being depicted on the shopping list. As subjects did not have to identify and remember the positions of the target places based on the shopping list during route planning, this control condition tested for the influence of spatial working memory for route planning performance. Results showed a strong performance increase in the control condition, emphasizing the prominent role of spatial working memory for route planning