Rational Normal Scrolls and the Defining Equations of Rees Algebras

Consider a height two ideal, $I$, which is minimally generated by $m$ homogeneous forms of degree $d$ in the polynomial ring $R=k[x,y]$. Suppose that one column in the homogeneous presenting matrix \f of $I$ has entries of degree $n$ and all of the other entries of \f are linear. We identify an explicit generating set for the ideal \Cal A which defines the Rees algebra \Cal R=R[It]; so \Cal R=S/\Cal A for the polynomial ring $S=R[T_1,...,T_m]$. We resolve \Cal R as an $S$-module and $I^s$ as an $R$-module, for all powers $s$. The proof uses the homogeneous coordinate ring, $A=S/H$, of a rational normal scroll, with H\subseteq \Cal A. The ideal \Cal AA is isomorphic to the $n^{\text{th}}$ symbolic power of a height one prime ideal $K$ of $A$. The ideal $K^{(n)}$ is generated by monomials. Whenever possible, we study $A/K^{(n)}$ in place of A/\Cal AA because the generators of $K^{(n)}$ are much less complicated then the generators of \Cal AA. We obtain a filtration of $K^{(n)}$ in which the factors are polynomial rings, hypersurface rings, or modules resolved by generalized Eagon-Northcott complexes. The generators of $I$ parameterize an algebraic curve \Cal C in projective $m-1$ space. The defining equations of the special fiber ring \Cal R/(x,y)\Cal R yield a solution of the implicitization problem for \Cal C.Comment: 48 page

Syzygies, finite length modules, and random curves

We apply the theory of Groebner bases to the computation of free resolutions over a polynomial ring, the defining equations of a canonically embedded curve, and the unirationality of the moduli space of curves of a fixed genus.Comment: 31 pages, This article consists of extended notes from lectures by the second author at the Joint Introductory Workshop: Cluster Algebras and Commutative Algebra at the Mathematical Science Research Institute (MSRI) in Berkeley, California, in Fall 201

The Dubrovin threefold of an algebraic curve

The solutions to the Kadomtsev-Petviashvili equation that arise from a fixed complex algebraic curve are parametrized by a threefold in a weighted projective space, which we name after Boris Dubrovin. Current methods from nonlinear algebra are applied to study parametrizations and defining ideals of Dubrovin threefolds. We highlight the dichotomy between transcendental representations and exact algebraic computations. Our main result on the algebraic side is a toric degeneration of the Dubrovin threefold into the product of the underlying canonical curve and a weighted projective plane.Comment: 28 page

Mechanism Singularities Revisited from an Algebraic Viewpoint

It has become obvious that certain singular phenomena cannot be explained by a mere investigation of the configuration space, defined as the solution set of the loop closure equations. For example, it was observed that a particular 6R linkage, constructed by a combination of two Goldberg 5R linkages, exhibits kinematic singularities at a smooth point in its configuration space. Such problems are addressed in this paper. To this end, an algebraic framework is used in which the constraints are formulated as polynomial equations using Study parameters. The algebraic object of study is the ideal generated by the constraint equations (the constraint ideal). Using basic tools from commutative algebra and algebraic geometry (primary decomposition, Hilbert's Nullstellensatz), the special phenomenon is related to the fact that the constraint ideal is not a radical ideal. With a primary decomposition of the constraint ideal, the associated prime ideal of one primary ideal contains strictly into the associated prime ideal of another primary ideal which also gives the smooth configuration curve. This analysis is extended to shaky and kinematotropic linkages, for which examples are presented.Comment: 6 figure

An explicit construction of ruled surfaces

The main goal of this paper is to give a general method to compute (via computer algebra systems) an explicit set of generators of the ideals of the projective embeddings of some ruled surfaces, namely projective line bundles over curves such that the fibres are embedded as smooth rational curves. Indeed, although the existence of the embeddings that we consider is well known, often in literature there are no explicit descriptions of the corresponding projective ideals. Such an explicit description allows to compute, besides all the syzygies, some of the important algebraic invariants of the surface, for instance the $k$-regularity, which are not always easy to compute by general formulae or by geometric arguments. An implementation of our algorithms and explicit examples for the computer algebra system Macaulay2 are included, so that anyone can use them for his own purposes.Comment: 22 page

Ideals of curves given by points

Let C be an irreducible projective curve of degree d in Pn(K), where K is an algebraically closed field, and let I be the associated homogeneous prime ideal. We wish to compute generators for I, assuming we are given sufficiently many points on the curve C. In particular if I can be generated by polynomials of degree at most m and we are given md + 1 points on C, then we can find a set of generators for I. We will show that a minimal set of generators of I can be constructed in polynomial time. Our constructions are completely independent of any notion of term ordering; this allows us the maximal freedom in performing our constructions in order to improve the numerical stability. We also summarize some classical results on bounds for the degrees of the generators of our ideal in terms of the degree and genus of the curve

A survey on Cox rings

We survey the construction of the Cox ring of an algebraic variety X and study the birational geometry of X when its Cox ring is finitely generated.Comment: 20 pages, 7 figure

Box Graphs and Resolutions I

Box graphs succinctly and comprehensively characterize singular fibers of elliptic fibrations in codimension two and three, as well as flop transitions connecting these, in terms of representation theoretic data. We develop a framework that provides a systematic map between a box graph and a crepant algebraic resolution of the singular elliptic fibration, thus allowing an explicit construction of the fibers from a singular Weierstrass or Tate model. The key tool is what we call a fiber face diagram, which shows the relevant information of a (partial) toric triangulation and allows the inclusion of more general algebraic blowups. We shown that each such diagram defines a sequence of weighted algebraic blowups, thus providing a realization of the fiber defined by the box graph in terms of an explicit resolution. We show this correspondence explicitly for the case of SU(5) by providing a map between box graphs and fiber faces, and thereby a sequence of algebraic resolutions of the Tate model, which realizes each of the box graphs.Comment: 44 pages, 53 figures, v2: typos fixed and clarifications adde

Extreme Rays of the Hankel Spectrahedra for Ternary Forms

Hankel spectrahedra are the dual convex cones to the cone of sums of squares of real polynomials, and we study them from the point of view of convex algebraic geometry. We show that the Zariski closure of the union of all extreme rays of Hankel spectrahedra for ternary forms is an irreducible variety of codimension 10. It is the variety of all Hankel (or middle Catalecticant) matrices of corank at least 4. We explicitly construct a rational extreme ray of maximal rank using the Cayley-Bacharach Theorem for plane curves. We work out the rank stratification of the semi-algebraic set of extreme rays of Hankel spectrahedra in the first three nontrivial cases d = 3, 4, 5. Dually, we get a characterisation of the algebraic boundary of the cone of sums of squares via projective duality theory, extending previous work of Blekherman, Hauenstein, Ottem, Ranestad, and Sturmfels.Comment: 25 pages, 3 figures. Comments welcome

The moduli space of even surfaces of general type with K^2 = 8, p_g = 4 and q = 0

We settle the first step for the classification of surfaces of general type with K^2 = 8, p_g = 4 and q = 0, classifying the even surfaces (K is 2-divisible). The first even surfaces of general type with $K^2=8$, $p_g=4$ and $q=0$ were found by Oliverio as complete intersections of bidegree (6,6) in a weighted projective space P(1,1,2,3,3). In this article we prove that the moduli space of even surfaces of general type with K^2 = 8, p_g = 4 and q = 0 consists of two 35 -dimensional irreducible components intersecting in a codimension one subset (the first of these components is the closure of the open set considered by Oliverio). For the surfaces in the second component the canonical models are always singular, hence we get a new example of generically nonreduced moduli spaces. Our result gives a posteriori a complete description of the half-canonical rings of the above even surfaces. The method of proof is, we believe, the most interesting part of the paper. After describing the graded ring of a cone we are able, combining the explicit description of some subsets of the moduli space, some deformation theoretic arguments, and finally some local algebra arguments, to describe the whole moduli space. This is the first time that the classification of a class of surfaces can only be done using moduli theory: up to now first the surfaces were classified, on the basis of some numerical inequalities, or other arguments, and later on the moduli spaces were investigated.Comment: 29 pages, the method of proof has been more carefully explaine