On the postulation of canonical curves in P3

We show that smooth curves in the same biliaison class on a hypersurface in $\mathbf{P}^3$ with ordinary singularities are linearly equivalent. We compute the invariants $h^0(\mathscr{I}_C(d))$, $h^1(\mathscr{I}_C(d))$ and $h^1(\mathscr{O}_C(d))$ of a curve $C$ on such a surface $X$ in terms of the cohomologies of divisors on the normalization of $X$. We then study general projections in $\mathbf{P}^3$ of curves lying on the rational normal scroll $S(a,b)\subset\mathbf{P}^{a+b+1}$. If we vary the curves in a linear system on $S(a,b)$ as well as the projections, we obtain a family of curves in $\mathbf{P}^3$. We compute the dimension of the space of deformations of these curves in $\mathbf{P}^3$ as well as the dimension of the family. We show that the difference is a linear function in $a$ and $b$ which does not depend on the linear system. Finally, we classify maximal rank curves on ruled cubic surfaces in $\mathbf{P}^3$. We prove that the general projections of all but finitely many classes of projectively normal curves on $S(1,2)\subset\mathbf{P}^4$ fail to have maximal rank in $\mathbf{P}^3$. These give infinitely many classes of counter-examples to a question of Hartshorne.

We present a new systematic method to construct Abelian functions on Jacobian varieties of plane, algebraic curves. The main tool used is a symmetric generalisation of the bilinear operator defined in the work of Baker and Hirota. We give explicit formulae for the multiple applications of the operators, use them to define infinite sequences of Abelian functions of a prescribed pole structure and deduce the key properties of these functions. We apply the theory on the two canonical curves of genus three, presenting new explicit examples of vector space bases of Abelian functions. These reveal previously unseen similarities between the theories of functions associated to curves of the same genus.

Decoding by rank-2 bundles over plane quartics

R. W. Brockett (1983) proposed to classify all finite dimensional estimation algebras. An affirmative solution to Brockett's problem will allow construction of all possible finite dimensional recursive filters from the Lie algebraic point of view. The concept of an estimation algebra with maximal rank was introduced by L. F. Tam et al. (1990). This is the most important general subclass of estimation algebras. S. S.-T. Yau and W.L. Chiou (1991) have already classified all maximal rank finite dimensional estimation algebras with state space dimension at most 2. Here, the case for state space dimension 3 is studied. >

We determine the Chow ring (with Q-coecients) of M6 by showing that all Chow classes are tautological. (In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective.) We stratify the moduli space into locally closed strata which are group quotients, and use a theorem of Vistoli to show that their Chow rings are generated by Chern classes of natural vector bundles. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus six case relies in addition on the particularly beautiful Brill{Noether theory in this case, and in particular on a rank ve vector bundle \relativizing a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the rank six vector bundle of quadrics cutting out the canonical curve.

A set of 2n points in P n − 1 is self-dual if it is invariant under the Gale transform. Motivated by Mukai’s work on canonical curves, Petrakiev showed that a general self-dual set of 14 points in P 6 arises as the intersection of the Grassmannian Gr ( 2 , 6 ) in its Plücker embedding in P 14 with a linear space of dimension 6. In this paper, we focus on the inverse problem of recovering such a linear space associated to a general self-dual set of points. We use numerical homotopy continuation to approach the problem and implement an algorithm in Julia to solve it. Along the way, we also implement the forward problem of slicing Grassmannians and use it to experimentally study the real solutions to this problem.

Introduction The notion of kernel-induced vector spaces is the cornerstone of kernel-based machine learning. Generalization of the traditional definition of a similarity metric plays a vital role in facilitating the analysis of complex and big data. It is often necessary to generalize the traditional Euclidean inner product to the more flexible and nonlinear inner products characterized by properly chosen kernel functions. The new inner product leads to a new distance metric, allowing the data analysis to be effectively performed in a higher-dimensional vector space. The topics addressed in this chapter are as follows. Section 2.2 introduces Mercer's fundamental theorem stating the necessary and sufficient condition for a function be a Mercer kernel. It will examine several prominent kernel functions, including the polynomial and Gaussian kernel functions. Section 2.3 introduces the notion of intrinsic space associated with a kernel function. The intrinsic space is so named because it is independent of the training dataset. The dimension of the space is denoted by J and will be referred to as the intrinsic degree . This degree, whether finite or infinite, dictates the training efficiency and computational cost. Section 2.4 introduces a finite-dimensional kernel-induced vector space, known as empirical space , which is jointly determined by the kernel function and the training dataset. The dimension of the empirical space is equal to the data size N . With the LSP condition, both the intrinsic-space and the kernelized learning model will be at our disposal.

In a recent series of short papers [2, 3, 41 I have been discussing the relationships of uniform convexity with certain other properties of normed vector spaces. In this paper I propose to discuss relationships between uniform convexity, factor spaces, and conjugate spaces. Because of the large number of special results which are needed for two dimensional spaces-and which are false in general-the paper will be divided into two parts: In Part I (??2-4) B is two dimensional; in Part II (??5-7) this restriction is removed and B may be any normed vector space. To recall the definition, B is said to be uniformly convex if there exists a function 6 such that 0 e, is a modulus of convexity for B. Note that if 51 is nowhere greater than 5, a modulus of convexity for B, then 5l is a modulus of convexity for B. Because of the pointwise nature of the definition of uniform convexity, it is clear that B is uniformly convex with modulus of convexity a if and only if all the two dimensional subspaces of B have the common modulus of convexity 6. One of the most useful results of this investigation (Theorem 5.5) is the complementary fact that B is uniformly convex if and only if all the two dimensional factor spaces of B have a common modulus of convexity. In the study of the effect of uniform convexity of B on the nature of the conjugate space B*, this result makes it possible to reduce the problem to the study of two dimensional spaces. In such a space B uniform convexity is equivalent to strict convexity: that is, a two (or finite) dimensional space is uniformly convex if and only if there does not exist a line segment of positive length all of whose points are of norm one. It has been observed [1, Footnote 13] that such a line segment on the unit sphere is equivalent to the existence of a sharp edge on the unit sphere in B*. The attempt to describe a sharp edge of the unit sphere in a finite dimensional space B in terms of the norm in that space leads to the condition that there exists a k > 0 such that for any e > 0 a pair of points bi and b2 exists such that II b -b2 11 k 11 bl-be 11 . Contradicting this suggests the following condition, a sort of dual concept to uniform convexity. A space B is said to be uniformly flattened if there exists a function v7 positive for 0 < e ? 2 such that lim.b.o 7(e) = 0 while (211 b + b2 11)/i b2 11 < 7(e) if J]b11 = JJ b1 = 1 and 11 bb2 ? || ; 375

Some necessary conditions for vector space partitions

Local linear dependence seen through duality II

In recent years, models of spatial relations, especially topological relations, have attracted much attention from the GIS community. In this paper, some basic topologic models for spatial entities in both vector and raster spaces are discussed. It has been suggested that, in vector space, an open set in 1-D space may not be an open set any more in 2-D and 3-D spaces. Similarly, an open set in 2-D vector space may also not be an open set any more in 3-D vector spaces. As a result, fundamental topological concepts such as boundary and interior are not valid any more when a lower dimensional spatial entity is embedded in higher dimensional space. For example, in 2-D, a line has no interior and the line itself (not its two end-points) forms a boundary. Failure to recognize this fundamental topological property will lead to topological paradox. It has also been stated that the topological models for raster entities are different in Z^{2} and R^{2}. There are different types of possible boundaries depending on the definition of adjacency or connectedness. If connectedness is not carefully defined, topological paradox may also occur. In raster space, the basic topological concept in vector space—connectedness—is implicitly inherited. This is why the topological properties of spatial entities can also be studied in raster space. Study of entities in raster (discrete) space could be a more efficient method than in vector space, as the expression of spatial entities in discrete space is more explicit than that in connected space.

Efficient Spectral-Chaos Methods for Uncertainty Quantification in Long-Time Response of Stochastic Dynamical Systems

The aim of this chapter is to give an account of the basic results which are needed to work with lattices, i.e. discrete subgroups of maximal rank in Euclidean vector spaces. Throughout the book, E stands for a real vector space of finite dimension n. From Section 1.2 onwards, the space is assumed to be Euclidean.

The second degree cohomology of finite orthogonal groups, II

With F q a finite field of characteristic p , let [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]( q ) be the category whose objects are functors from finite dimensional F q -vector spaces to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]-vector spaces. Friedlander and Suslin have introduced a category [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] of "strict polynomial functors" which has the same relationship to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /]( q ) that the category of rational GL m -modules has to the category of GL m ( F q )-modules. Our main theorem says that, for all finite objects F, G ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i" /], and all s , the natural restriction map from [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="06i" /] ( F ( k ) , G ( k ) ) to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="07i" /] ( F, G ) is an isomorphism for all large enough k and q . Here F ( k ) denotes F twisted by the Frobenius k times. This combines with an analogous theorem of Cline, Parshall, Scott, and van der Kallen to show that, for all finite F, G ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="08i" /], and all s , evaluation on an m dimensional vector space V m induces an isomorphism from [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="09i" /] ( F, G ) to Ext s GL m ( F q ) ( F ( V m ), F ( V m )) for all large enough m and q . Thus group cohomology of the finite general linear groups has often been identified with MacLane (or Topological Hochschild) cohomology.