Equations of Parametric Surfaces with Base Points via Syzygies
William Adkins, J. William Hoffman, Hao Hao Wang
备注：
22 pages. Revised version adds proofs that were originally quoted from Hoffman and Wang (arxiv.,org/abs/math.AG/0305125). The paper of Hoffman and Wang has now been withdrawn
Let $S$ be a parametric surface in $\proj{3}$ given as the image of $ϕ: \proj{1} \times \proj{1} \to \proj{3}$. This paper will show that the use of syzygies in the form of a combination of moving planes and moving quadrics provides a valid method for finding the implicit equation of $S$ when certain base points are present. This work extends the algorithm provided by Cox for when $ϕ$ has no base points, and it is an analogous to some of the results of Busé, Cox and D'Andrea for the case when $ϕ: \proj{2} \to \proj{3}$ has base points.
The revised version has two additional references and a shorter proof of Proposition 5.7. This version also makes numerous small changes and has an appendix containing a proof of the degree formula for a parametrized surface.
We describe the equations and Gröbner bases of some degenerate K3 surfaces associated to rational normal scrolls. These K3 surfaces are members of a class of interesting singular projective varieties we call correspondence scrolls. The ideals of these surfaces are nested in a simple way that allows us to analyze them inductively. We describe explicit Gröbner bases and syzygies for these objects over the integers and this lets us treat them in all characteristics simultaneously.
Based on the theory of an infinitesimal Newton-Okounkov body, we extend the results of Lazarsfeld-Pareschi-Popa on abelian surfaces. Moreover, we show that the higher syzygies of $(X,L)$ are completely determined by its Seshadri constant when $L^{2}$ is large. As an application, we improve the existing lower bound of $(L^{2})$ for higher syzygies of a polarized abelian surface $(X,L)$.
Let X be a smooth complex projective variety of dimension n and let A be an ample and basepoint free divisor. We prove $K_X+mA$ satisfies property $N_p$ for $m\geqslant n+1+p$. We also show the graded ring of sections $R(X, K_X+mA)$ is Koszul for $m\geqslant n+2$.
We prove new results on projective normality, normal presentation and higher syzygies for a surface of general type $X$ embedded by adjoint line bundles $L_r = K + rB$, where $B$ is a base point free, ample line bundle. Our main results determine the $r$ for which $L_r$ has $N_p$ property. In corollaries, we will relate the bounds on $r$ to the regularity of $B$. Examples in the last section show that several results are optimal.
Let $X$ be flat scheme over $\mathbb{Z}$ such that its base change, $X_p$, to $\bar{\mathbb{F}}_p$ is Frobenius split for all primes $p$. Let $G$ be a reductive group scheme over $\mathbb{Z}$ acting on $X$. In this paper, we prove a result on the $N_p$ property for line bundles on GIT quotients of $X_{\mathbb{C}}$ for the action of $G_{\mathbb{C}}$. We apply our result to the special cases of (1) an action of a finite group on the projective space and (2) the action of a maximal torus on the flag variety of type $A_n$.
Based on a recent result of Voisin [2001] we describe the last nonzero syzygy space in the linear strand of a canonical curve C of even genus g=2k lying on a K3 surface, as the ambient space of a k-2-uple embedded P^{k+1}. Furthermore the geometric syzygies constructed by Green and Lazarsfeld [1984] from g^1_{k+1}'s form a non degenerate configuration of finitely many rational normal curves on this P^{k+1}. This proves a natural generalization of Green's conjecture [1984], namely that the geometric syzygies should span the space of all syzygies, in this case.
The goal of this article is to study the equations and syzygies of embeddings of rational surfaces and certain Fano varieties. Given a rational surface X and an ample and base-point-free line bundle L on X, we give an optimal numerical criterion for L to satisfy property Np. This criterion turns out to be a characterization of property Np if X is anticanonical. We also prove syzygy results for adjunction bundles and a Reider type theorem for higher syzygies. For certain Fano varieties we also prove results on very ampleness and higher syzygies.
We derive the implicit equations for certain parametric surfaces in three-dimensional projective space termed tensor product surfaces. Our method computes the implicit equation for such a surface based on the knowledge of the syzygies of the base point locus of the parametrization by means of constructing an explicit virtual projective resolution.
We show that the implicit equation of a surface in 3-dimensional projective space parametrized by bi-homogeneous polynomials of bi-degree (d,d), for a given positive integer d, can be represented and computed from the linear syzygies of its parametrization if the base points are isolated and form locally a complete intersection.
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.
One proves a far-reaching upper bound for the degree of a generically finite rational map between projective varieties over a base field of arbitrary characteristic. The bound is expressed as a product of certain degrees that appear naturally by considering the Rees algebra (blowup) of the base ideal defining the map. Several special cases are obtained as consequences, some of which cover and extend previous results in the literature.
A graded K-algebra R has property N_p if it is generated in degree 1, has relations in degree 2 and the syzygies of order less or equal to p on the relations are linear. The Green-Lazarsfeld index of R is the largest p such that it satisfies the property N_p. Our main results assert that (under a mild assumption on the base field K) the c-th Veronese subring of a polynomial ring has Green-Lazarsfeld index greater or equal to c+1. The same conclusion also holds for an arbitrary standard graded algebra, provided c is sufficiently large.
First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field. Special case of the result is a finite presentation of the algebra of multisymmetric polynomials. Reducedness of the invariant commuting scheme is proved as a by-product. The algebra of multisymmetric polynomials over an arbitrary base ring is revisited.
We present a construction explaining the existence of (unexpected) curves of degree $d+k$, passing through a set $Z$ of points on $\mathbb{P}^2$, and having a generic point $P$ of multiplicity $d$. The construction is based on the syzygies of the $k$-th powers of Jacobian of the product of lines dual to the points of $Z$. We prove also a result characterizing the unexpectedness of the curves via splitting type of the bundle of these syzygies retricted to the line dual to $P$.
We investigate the application of syzygies for efficiently computing (finite) Pommaret bases. For this purpose, we first describe a non-trivial variant of Gerdt's algorithm to construct an involutive basis for the input ideal as well as an involutive basis for the syzygy module of the output basis. Then we apply this new algorithm in the context of Seiler's method to transform a given ideal into quasi stable position to ensure the existence of a finite Pommaret basis. This new approach allows us to avoid superfluous reductions in the iterative computation of Janet bases required by this method…
We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner bases for each higher syzygy module. In each case the given minimal Gröbner bases is also a minimal generating set. The Betti numbers of the binomial ideal and its natural initial ideal coincide and they correspond to the number of 'connected flags' in the graph. In particular the Betti numbers are independent of the characteristic of the base field. For complete graphs the problem was previously studied b…
Let L be a normally generated line bundle on X; we say L satisfies property N_p (notation after Mark Green) if the matrices in the free resolution of R (the homogeneous coordinate ring of X) over S (the homogeneous coordinate ring of the projective space corresponding to the complete linear series |L|) have linear entries until the p-th stage. In this article we prove the following result: Let X be an elliptic ruled surface and let L be a product of p+1 base point free and ample line bundles on X. Then L satisfies property N_p. In particular we prove that numerical classes of all divisors whic…
Parameterized algebraic curves and surfaces are widely used in geometric modeling and their manipulation is an important task in the processing of geometric models. In particular, the determination of the intersection loci between points, pieces of parameterized algebraic curves and pieces of algebraic surfaces is a key problem in this context. In this paper, we survey recent methods based on syzygies and blowup algebras for computing the image and the finite fibers of a curve or surface parameterization, more generally of a rational map. Conceptually, the main idea is to use elimination matri…
We show that the method of moving quadrics for implicitizing surfaces in P^3 applies in certain cases where base points are present. However, if the ideal defined by the parametrization is saturated, then this method rarely applies. Instead, we show that when the base points are a local complete intersection, the implicit equation can be computed as the resultant of the first syzygies.
In this paper we present an algorithm for computing a matrix representation for a surface in P^3 parametrized over a 2-dimensional toric variety T. This algorithm follows the ideas of [Botbol-Dickenstein-Dohm-09] and it was implemented in Macaulay2. We showed in [BDD09] that such a surface can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection, and in [Botbol-09] we generalized this to the case where the base locus is not necessarily a local complete intersection. The key point consists in exploiting the sparse structu…
A tensor product surface $\mathscr{S}$ is an algebraic surface that is defined as the closure of the image of a rational map $ϕ$ from $\mathbb{P}^1\times \mathbb{P}^1$ to $\mathbb{P}^3$. We provide new determinantal representations of $\mathscr{S}$ under the assumptions that $ϕ$ is generically injective and its base points are finitely many and locally complete intersections. These determinantal representations are matrices that are built from the coefficients of linear relations (syzygies) and quadratic relations of the bihomogeneous polynomials defining $ϕ$. Our approach relies on a formaliz…
A parameterized surface can be represented as a projection from a certain toric surface. This generalizes the classical homogeneous and bihomogeneous parameterizations. We extend to the toric case two methods for computing the implicit equation of such a rational parameterized surface. The first approach uses resultant matrices and gives an exact determinantal formula for the implicit equation if the parameterization has no base points. In the case the base points are isolated local complete intersections, we show that the implicit equation can still be recovered by computing any non-zero maxi…
