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.
The aim of this paper is to get some results about ruled surfaces which configure a projective theory of scrolls and ruled surfaces. Our ideas follow the viewpoint of Corrado Segre, but we employ the contemporaneous language of locally free sheaves. The results complete the exposition given by R. Hartshorne and they have not appeared before in the contemporaneous literature.
Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points correspond to smooth irreducible non-degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. We use families of curves on cones to show that under certain numerical assumptions for $d$, $g$ and $r$, the scheme $\mathcal{I}_{d,g,r}$ acquires generically smooth components whose general points correspond to curves that are double covers of irrational curves. In particular, in the case $ρ(d,g,r) := g-(r+1)(g-d+r) \geq 0$ we construct explicitly a regular component that is different …
We prove the existence of canonical scrolls; that is, scrolls playing the role of canonical curves. First of all, they provide the geometrical version of Riemann Roch Teorem: any special scroll is the projection of a canonical scroll and they allow to understand the classification of special scrolls in P3. Canonical scrolls correspond to the projective model of canonical geometrically ruled surfaces over a smooth curve. We also prove that the generic canonical scroll is projectively normal except in the hyperelliptic case and for very particular cases in the nonhyperelliptic situation.
We show that the generic fiber of a family of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking the base, such a family actually factors through an $\mathbb{A}^{1}$-fibration over a certain scheme, induced by the MRC-fibration of a relative smooth projective model of the family. For affine threefolds fibered by irrational $\mathbb{A}^{1}$-ruled surfaces, this induced $\mathbb{A}^{1}$-fib…
We compute the degree of the variety parametrizing rational ruled surfaces of degree d in the projective space by relating the problem to Gromov-Witten invariants and Quantum cohomology.
We investigate the structure of geometrically ruled surfaces whose anti-canonical class is big. As an application we show that the Picard group of a normal projective surface whose anti-canonical class is nef and big is a free abelian group of finite rank.
We study Fourier--Mukai partners of elliptic ruled surfaces. We also describe the autoequivalence group of the derived categories of ruled surfaces with an elliptic fibration, by using \cite{Ue15}.
In this paper, we give a complete description of the deformation classes of real structures on minimal ruled surfaces. In particular, we show that these classes are determined by the topology of the real structure, which means that real minimal ruled surfaces are quasi-simple. As an intermediate result, we obtain the classification, up to conjugation, of real structures on decomposable ruled surfaces.
We classify the singularities of a surface ruled by conics: they are rational double points of type $A_n$ or $D_n$. This is proved by showing that they arise from a precise series of blow-ups of a suitable surface geometrically ruled by conics. We determine also the family of such surfaces which are birational models of a given surface ruled by conics and obtained in a "minimal way" from it.
In arXive:0705.3912 we studied triple-point defective very ample linear systems on regular surfaces, and we showed that they can only exist if the surface is ruled. In the present paper we show that we can drop the regularity assumption, and we classify the triple-point defective very ample linear systems on ruled surfaces.
We give counterexamples to the Kawamata-Viehweg vanishing theorem on ruled surfaces in positive characteristic, and prove that if there is a counterexample to the Kawamata-Viehweg vanishing theorem on a geometrically ruled surface f:X-->C, then either C is a Tango curve or all of sections of f are ample.
The ruled surfaces, i.e., surfaces generated by one parametric set of lines, are widely used in the~field of applied geometry. An~isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some fixed vector. Choosing an angle equal to $π/2$ we obtain a special instance of a~isophote -- the so called contour curve. While contours on rational ruled surfaces are rational curves, this is no longer true for the isophotes. Hence we will provide a formula for their genus. Moreover we will show that the only surfaces with a~rational generic contour are just …
In this paper, we compare the moduli spaces of rank-3 vector bundles stable with respect to different ample divisors over rational ruled surfaces. We also discuss the irreducibility, unirationality, and rationality of these moduli spaces.
Chen, Sederberg, and Zheng introduced the notion of a $μ$-basis for a rational ruled surface in Chen et al. (2001) and showed that its resultant is the implicit equation of the surface, if the parametrization is generically injective. We generalize this result to the case of an arbitrary parametrization. We also give a new proof for the corresponding theorem in the curve case and treat the reparametrization problem for curves and ruled surfaces.
We give a explicit description of gluing stability conditions on ruled surfaces by introducing gluing perversity. Moreover, we describe a destabilizing wall of skyscraper sheaves on ruled surfaces by deformation of stability conditions glued from $\widetilde{GL^{+}}(2,\mathbb{R})$-translates of the standard stability condition on the base curve.
We study the Seshadri constants on geometrically ruled surfaces. The unstable case is completely solved. Moreover, we give some bounds for the stable case. We apply these results to compute the Seshadri constant of the rational and elliptic ruled surfaces. Both cases are completely determined. The elliptic case provides an interesting picture of how particular is the behavior of the Seshadri constants.
It is proved that a smooth rational surface in projective four-space, which is ruled by cubics or quartics has degree at most 12. It is also proved that a smooth rational surface in projective four-space which is the image of Fn by a linear system with "simple base points" (see text) has degree at most 12.
In this work we use arithmetic, geometric, and combinatorial techniques to compute the cohomology of Weil divisors of a special class of normal surfaces, the so-called rational ruled toric surfaces. These computations are used to study the topology of cyclic coverings of such surfaces ramified along Q-normal crossing divisors.
Very ampleness criteria for rank 2 vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.
We study the order of automorphism groups of cyclic covering fibrations of a ruled surface. Arakawa and later Chen studied it for hyperelliptic fibrations and gave the upper bound. The purpose of present paper is to pursue the analog for cyclic covering fibrations.
We give an algorithm to classify singular fibers of finite cyclic covering fibrations of a ruled surface by using singularity diagrams. As the first application, we classify all fibers of 3-cyclic covering fibrations of genus 4 of a ruled surface and show that the signature of a complex surface with this fibration is non-positive by computing the local signature for any fiber. As the second application, we classify all fibers of hyperelliptic fibrations of genus 3 into 12 types according to the Horikawa index. We also prove that finite cyclic covering fibrations of a ruled surface have no mult…
The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta - Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be considered as a weak form of semistability. We establish a Bogomolov type inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section. This gives an affermative answer in this case to a problem posed by Th. Peternell. Concerning ruled surfaces with a negative section, we prove a a similar result fo…
We prove that the morphism that maps a rational ruled surface to its singular locus is genericaly injective modulo isomophism and duality. We also calculate the dimension and the degre of its image.
In this paper we give the projective generation of congruences of order 1 of r-dimensional projective spaces in P^N from their focal loci. In a natural way, this construction shows that the corresponding surfaces in the grassmannian are the Veronese surface, and rational ruled surfaces eventually with singularities. We characterize when these surfaces are smooth, recovering and generalizing a Ziv Ran's result.
We study moduli spaces $M_X(r,c_1,c_2)$ parametrizing slope semistable vector bundles of rank $r$ and fixed Chern classes $c_1, c_2$ on a ruled surface whose base is a rational nodal curve. We show that under certain conditions, these moduli spaces are irreducible, smooth and rational (when non-empty). We also prove that they are non-empty in some cases. We show that for a rational ruled surface defined over real numbers, the moduli space $M_X(r,c_1,c_2)$ is rational as a variety defined over $\mathbb R$.
Let $X$ be a surface and let $L$ be an ample line bundle on $X$. We first obtain a lower bound for the Seshadri constant $\varepsilon(X,L,r)$, when $r \ge 3$. We then assume that $X$ is a ruled surface and study Seshadri constants on $X$ in greater detail. We also make precise computations of Seshadri constants on $X$ in some cases.
Let $X$ be a smooth projective surface and let $\mathcal{C}$ be an arrangement of curves on $X$. The Harbourne constant of $\mathcal{C}$ was defined as a way to investigate the occurrence of curves of negative self-intersection on blow ups of $X$. This is related to the bounded negativity conjecture which predicts that the self-intersection number of all reduced curves on a surface is bounded below by a constant. We consider a geometrically ruled surface $X$ over a smooth curve and give lower bounds for the Harbourne constants of transversal arrangements of curves on $X$.
Let $S$ be a ruled surface inside a smooth threefold $W$ and let $E$ be a vector bundle on a formal neighborhood of $S.$ We find minimal conditions under which the local moduli space of $E$ is finite dimensional and smooth. Moreover, we show that $E$ is a flat limit of a flat family of vector bundles whose general element we describe explicitly.
Atiyah classifies vector bundles on elliptic curves $E$ over an algebraically closed field of any characteristic. On the other hand, a rank $2$ vector bundle on $E$ defines a surface $S$ with a $\mathbb{P}^1$-bundle structure on $E$. We study when $S$ has an elliptic fibration according to the Atiyah's classification, and what kinds of singular fibers appear.
A conjecture of Morel asserts that the sheaf of $\mathbb A^1$-connected components of a space is $\mathbb A^1$-invariant. Using purely algebro-geometric methods, we determine the sheaf of $\mathbb A^1$-connected components of a smooth projective surface, which is birationally ruled over a curve of genus $>0$. As a consequence, we show that Morel's conjecture holds for all smooth projective surfaces over an algebraically closed field of characteristic $0$.
Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on S. We prove that the full moduli stack for this problem is representable by a scheme in some cases. Moreover, the moduli stack admits an action by the group ${\mathbf C}^*$, and we determine its fixed-point set, which leads in some special cases to explicit formulas for the rational homology of the moduli space.
The paper is a generalization of a result of I. Dolgachev, M. Mendes Lopes, and R. Pardini. We prove that a smooth projective complex surface $X$, not necessarily minimal, contains $h^{1,1}(X)-1$ disjoint $(-2)$-curves if and only if $X$ is isomorphic to a minimal rational ruled surface ${\bf F}_2$ or ${\bf P}^2$ or a fake projective plane. We also describe smooth projective complex surfaces $X$ with $h^{1,1}(X)-2$ disjoint $(-2)$-curves.
A smooth family $φ:\mathcal V\to S$ of surfaces will be called {\em completable} if there is a logarithmic deformation $(\bar {\mathcal V},{\mathcal D})$ over $S$ so that ${\mathcal V}=\bar{\mathcal V}\backslash {\mathcal D}$. Two smooth surfaces $V$ and $V'$ are said to be deformations of each other if there is a completable flat family ${\mathcal V}\to S$ of smooth surfaces over a connected base so that $V$ and $V'$ are fibers over suitable points $s,s'\in S$. This relation generates an equivalence relation called {\em deformation equivalence}. In this paper we give a complete combinatorial …
We study ruled orders. These arise naturally in the Mori program for orders on projective surfaces and morally speaking are orders on a ruled surface ramified on a bisection and possibly some fibres. We describe fibres of a ruled order and show they are in some sense rational. We also determine the Hilbert scheme of rational curves and hence the corresponding non-commutative Mori contraction. This gives strong evidence that ruled orders are examples of the non-commutative ruled surfaces introduced by Van den Bergh.
We provide algorithms to reconstruct rational ruled surfaces in three-dimensional projective space from the `apparent contour' of a single projection to the projective plane. We deal with the case of tangent developables and of general projections to $\mathbb{p}^3$ of rational normal scrolls. In the first case, we use the fact that every such surface is the projection of the tangent developable of a rational normal curve, while in the second we start by reconstructing the rational normal scroll. In both instances we then reconstruct the correct projection to $\mathbb{p}^3$ of these surfaces by…
We present a method for computing all the symmetries of a rational ruled surface defined by a rational parametrization which works directly in parametric rational form, i.e. without computing or making use of the implicit equation of the surface. The method proceeds by translating the problem into the parameter space, and relies on polynomial system solving. If we want all the symmetries of the surface, including rotational symmetries, we need to deal with polynomial systems in four variables; if we are only interested in involutions (e.g. central symmetries, axial symmetries, reflections in a…
We consider the following question: for which invariants $g$ and $e$ is there a geometrically ruled surface $S \rightarrow C$ over a curve $C$ of genus $g$ with invariant $e$ such that $S$ is the support of an Ulrich line bundle with respect to a very ample line bundle? A surprising relation between the existence of certain proper Theta divisors on some moduli spaces of vector bundles on $C$ with the existence of Ulrich line bundles on $S$ will be the key to completely solve the above question. The relation is realized by translating the vanishing conditions characterizing Ulrich line bundles …
We consider the Quot scheme, R_{d}, compactifying the space of degree d maps from the projective line to the Grassmannian of lines. We give an algorithm for computing the degree of R_{d} under a "generalized Plücker embedding", this is a certain Gromov-Witten invariant. The approach is to apply the Atiyah-Bott localization formula for the natural C^{*}-action on R_{d}. These numbers can be obtained directly from Vafa-Intriligator formula.
We study higher syzygies of a ruled surface $X$ over a curve of genus $g$ with the numerical invariant $e$. Let $L \in {Pic}X$ be a line bundle in the numerical class of $aC_0 +bf$. We prove that for $0 \leq e \leq g-3$, $L$ satisfies property $N_p$ if $a \geq p+2$ and $b-ae \geq 3g-1-e+p$ and for $e \geq g-2$, $L$ satisfies property $N_p$ if $a \geq p+2$ and $b-ae\geq 2g+1+p$. By using these facts, we obtain Mukai type results. For ample line bundles $A_i$, we show that $K_X + A_1 + ... + A_q$ satisfies property $N_p$ when $0 \leq e < \frac{g-3}{2}$ and $q \geq g-2e+1 +p$ or when $e \geq \fra…
In this article, we give a description of the closed cone of curves of the projective bundle $\mathbb{P}(E)$ over a smooth projective variety $X$. Using duality, we then calculate the nef cone of divisors in $\mathbb{P}(E)$ over some special surfaces $X$ and for some special bundles on $X$. As an application, we also calculate the Seshadri constants of semistable ample vector bundles with vanishing discriminant on some special ruled surfaces at special points.
Motivated by a number of recent investigations, we define and investigate the various properties of the ruled surfaces depend on three dimensional Lie groups with a bi-variant metric. We give useful results involving the characterizations of these ruled surfaces. Some special ruled surfaces such as normal surface, binormal surface, tangent developable surface, rectifying developable surface and Darboux developable surface are worked. From those applications, we make use of such a work to interpret the Gaussian, mean curvatures of these surfaces and geodesic, normal curvature and geodesic torsi…
A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus $g\ge 1$ is itself a ruled surface over a curve of genus $g$. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in case $g\ge 2$.
We prove that an analogue of Jordan's theorem on finite subgroups of general linear groups holds for the groups of biregular automorphisms of elliptic ruled surfaces. This gives a positive answer to a question of Vladimir L. Popov.
We explicitly describe the $\mathbb A^1$-chain homotopy classes of morphisms from a smooth henselian local scheme into a smooth projective surface, which is birationally ruled over a curve of genus $> 0$. We consequently determine the sheaf of naive $\mathbb A^1$-connected components of such a surface and show that it does not agree with the sheaf of its genuine $\mathbb A^1$-connected components when the surface is not a minimal model. However, the sections of the sheaves of both naive and genuine $\mathbb A^1$-connected components over schemes of dimension $\leq 1$ agree. As a consequence, w…
