Given n general points p 1 , p 2 , . . . , p n ∈ P r , it is natural to ask when there exists a curve C ⊂ P r , of degree d and genus g, passing through p 1 , p 2 , . . . , p n . In this paper, we give a complete answer to this question for curves C with nonspecial hyperplane section. This result is a consequence of our main theorem, which states that the normal bundle N C of a general nonspecial curve of degree d and genus g in P r (with d ≥ g + r) has the property of interpolation (i.e. that for a general effective divisor D of any degree on C, either H 0 (N C (−D)) = 0 or H 1 (N C (−D)) = 0), with exactly three exceptions.
of the paper. In §2 we introduce the toric dictionary and, following Gonzalez-Sprinberg [6], translate the question into convex geometry. In §3 we summarize the effect of the iterated Nash blow-up, in the toric case, using the notion of a resolution tree. In §4 we spell out what happens in the 2-dimensional toric case in terms of continued fractions. In §5 we digress briefly on the classification of quasi-smooth affine toric varieties, those corresponding to simplicial cones in the toric dictionary. Then in §6 we give an account of our computer investigations.Notation and conventions. We slightly abuse terminology in two ways. First, as we are concerned with the normalized Nash blow-up throughout, we refer to it simply as the Nash blow-up. Second, as we are concerned with rational polyhedra and rational polyhedral cones throughout, we refer to them simply as polyhedra and cones. We denote the natural numbers, including 0, by Z + , and we likewise denote the nonnegative rational numbers, including 0, by Q + . We denote the span of v 1 , . . . , v k with coefficients in S by S v 1 , . . . , v k . Thus, for example, the first quadrant in Q 2 is denoted Q + e 1 , e 2 .Acknowledgements. The very helpful advice provided by Kevin Purbhoo is gratefully acknowledged. We also thank Jeffrey Lagarias for an inspiring conversation, Dave Bayer for recommending the use of 4ti2, and Sam Payne and Greg Smith for informing us of their parallel work on the subject.
Motivated by the result of Rankin for representations of integers as sums of squares, we use a decomposition of a modular form into a particular Eisenstein series and a cusp form to show that the number of ways of representing a positive integer n as the sum of k triangular numbers is asymptotically equivalent to the modified divisor function σ 2k−1 (2n + k). 2 for 0 (4) to study the functions r k (n). Ono, Robins, and Wahl [1995] defined an analogous modular form to study triangular numbers. We begin by defining triangular numbers.
Abstract. We use the nudged elastic band method from computational chemistry to analyze high-dimensional data. Our approach is inspired by Morse theory, and as output we produce an increasing sequence of small cell complexes modeling the dense regions of the data. We test the method on data sets arising in social networks and in image processing. Furthermore, we apply the method to identify new topological structure in a data set of optical flow patches.
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