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.
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.
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.
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 …
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.
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 $\mathbb R$.
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.
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.
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.
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 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 present an algorithm that covers any given rational ruled surface with two rational parametrizations. In addition, we present an algorithm that transforms any rational surface parametrization into a new rational surface parametrization without affine base points and such that the degree of the corresponding maps is preserved.
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 $X$.
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…
In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety $V$ which is covered by lines. The main technical result used to achieve this is an upper bound on the number of rational points of bounded height on a line. This upper bound is such that it can be easily controlled as the line varies, and hence is used to sum the counting functions of the lines which cover the original variety $V$.
Let $Σ$ be a smooth projective surface, let $f' : S' \to Σ$ be a double cover of $Σ$ and let $μ: S \to S'$ be the canonical resolution. Put $f = f'\circμ$. An irreducible curve $C$ on $Σ$ is said to be a splitting curve with respect to $f$ if $f^*C$ is of the form $C^+ + C^- + E$, where $C^- = σ_f^*C^+$, $σ_f$ being the covering transformation of $f$ and all irreducible components of $E$ are contained in the exceptional set of $μ$. In this article, we show that a kind of "reciprocity" of splitting curves holds for a certain pair of curves on rational ruled surfaces. As an application, we consi…
The ruled surface is a typical modeling surface in computer aided geometric design. It is usually given in the standard parametric form. However, it can also be in the forms than the standard one. For these forms, it is necessary to determine and find the standard form. In this paper, we present algorithms to determine whether a given implicit surface is a rational ruled surface. A parametrization of the surface is computed for the affirmative case. We also consider the parametric situation. More precisely, after a given rational parametric surface is determined as a ruled one, we reparameteri…
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 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.
We study rationality properties of smooth complete intersections of three quadrics in $\mathbb{P}^7$. We exhibit a smooth family of such intersections with both rational and non-rational fibers.
We study properties of rational curves on complete intersections in positive characteristic. It has long been known that in characteristic 0, smooth Calabi-Yau and general type varieties are not uniruled. In positive characteristic, however, there are well-known counterexamples to this statement. We will show that nevertheless, a \emph{general} Calabi-Yau or general type complete intersection in projective space is not uniruled. We will also show that the space of complete intersections of degree $(d_1, \cdots, d_k)$ containing a rational curve has codimension at least $\sum_{i=1}^k d_i - 2n +…
Complete intersections inside rational homogeneous varieties provide interesting examples of Fano manifolds. For example, if $X = \cap_{i=1}^r D_i \subset G/P$ is a general complete intersection of $r$ ample divisors such that $K_{G/P}^* \otimes \mathcal{O}_{G/P}(-\sum_i D_i)$ is ample, then $X$ is Fano. We first classify these Fano complete intersections which are locally rigid. It turns out that most of them are hyperplane sections. We then classify general hyperplane sections which are quasi-homogeneous.
In this paper, we consider rational cuspidal plane curves having at least three cusps. We give an upper bound of the self-intersection number of the proper transforms of such curves via the minimal embedded resolution of the cusps. For a curve having exactly three cusps, we show that the self-intersection number is equal to the bound if and only if the curve coincides with the quartic curve having three cusps.
A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be rationally connected themselves. We prove that smooth, complex, weighted complete intersections of low enough degree are rationally simply connected. This result has strong arithmetic implications for weighted complete intersections defined over the function field of a smooth, complex curve. Namely, it implies that these varieties satisfy weak approximation at all p…
In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface $\fn$. We compute the number of such curves through the appropriate number of fixed general points on $\fn$, and the number of such curves which are irreducible. These numbers are known as Severi degrees; they are the degrees of unions of components of the Hilbert scheme. As (i) $\fn$ can be deformed to $\eff_{n+2}$, (ii) the Gromov-Witten invariants are deformation-invariant, and (iii) the Gromov-Witten invariants of $\eff_0$ and $\eff_1$ are enumerative, Theorem \ref{irecursion} com…
Let $X \subset \mathbb{P}^n$ be a general Fano complete intersection of type $(d_1,\dots, d_k)$. If at least one $d_i$ is greater than $2$, we show that $X$ contains rational curves of degree $e \leq n$ with balanced normal bundle. If all $d_i$ are $2$ and $n\geq 2k+1$, we show that $X$ contains rational curves of degree $e \leq n-1$ with balanced normal bundle. As an application, we prove a stronger version of the theorem of Z. Tian \cite{Tian}, Q. Chen and Y. Zhu \cite{ChenZhu} that $X$ is separably rationally connected by exhibiting very free rational curves in $X$ of optimal degrees.
We prove the following results. If $X_3$ is a generic complete intersection Calabi-Yau 3-fold, (1) then for each natural number $d$ there exists a rational map \par\hspace{1 cc} $c\in Hom_{bir}(\mathbf P^1, X_3)$ of $deg(c(\mathbf P^1))=d$, (2) further more all such $c$ are immersions satisfying \begin{equation} N_{c(\mathbf P^1)/ X_3}\simeq \mathcal O_{\mathbf P^1}(-1)\oplus \mathcal O_{\mathbf P^1}(-1).
We prove that the space of smooth rational curves of degree $e$ in a general complete intersection of multidegree $(d_1, ..., d_m)$ in $\PP^n$ is irreducible of the expected dimension if $\sum_{i=1}^m d_i <\frac{2n}{3}$ and $n$ is large enough. This generalizes the results of Harris, Roth and Starr \cite{hrs}, and is achieved by proving that the space of conics passing through any point of a general complete intersection has constant dimension if $\sum_{i=1}^m d_i$ is small compared to $n$.
Let X be a smooth complex projective surface. We prove that for any sufficiently big m there exists a rational dominant map f from X into a complex rational ruled surface Y, such that f is generically finite of degree m and has monodromy the alternating group Am.
This paper is to give some concrete examples of the general fibers of the evaluation map of some Kontsevich mapping spaces parametrize low degree rational curves on low degree complete intersection varieties. We prove these examples are Fano or Calabi-Yau vareities of low dimension, e.g, surface and threefold. In usual, the "moduli" fibers are constructed in an abstract way, but here we give a down-to-earth interpretation to some of these fibers from the perspective of projective geometry.
For a smooth complete intersection X, we consider a general fiber \mathbb{F} of the evaluation map ev of Kontsevich moduli space \bar{M}_{0,m}(X,m)\rightarrow X^m and the forgetful functor F : \mathbb{F} \rightarrow \bar{M}_{0,m}. We prove that a general fiber of the map $F$ is a smooth complete intersection if $X$ is of low degree. The result sheds some light on the arithmetic and geometry of Fano complete intersections.
Using the Grothendieck-Lefschetz theory (see \cite{[SGA2]}) we prove a criterion to deduce that certain subvarieties of $\mathbb P^n$ of dimension $\geq 2$ are not set-theoretic complete intersections (see Theorem 1 of the Introduction). As applications we give a number of relevant examples. In the last part of the paper we prove that the arithmetic rank of a rational normal $d$-dimensional scroll $S_{n_1,...,n_d}$ in $\mathbb P^N$ is $N-2$, by producing an explicit set of $N-2$ homogeneous equations which define these scrolls set-theoretically (see Theorem 2 of the Introduction).
The Hurwitz space is a compactification of the space of rational functions of a given degree. The Lyashko-Looijenga map assigns to a rational function the set of its critical values. It is known that the number of ramified coverings of CP^1 by CP^1 with prescribed ramification points and ramification types is related to the degree of the Lyashko--Looijenga map on various strata of the Hurwitz space. Here we explain how the degree of the Lyashko-Looijenga map is related to the intersection theory on this space. We describe the cohomology algebra of the Hurwitz space and prove several relations …
We give the first evidence for a conjecture that a general, index-one, Fano hypersurface is not unirational: (i) a general point of the hypersurface is contained in no rational surface ruled, roughly, by low-degree rational curves, and (ii) a general point is contained in no image of a Del Pezzo surface.
In a previous paper the authors develop an intersection theory for subspaces of rational functions on an algebraic variety X over complex numbers. In this note, we first extend this intersection theory to an arbitrary algebraically closed ground field. Secondly we give an isomorphism between the group of b-divisors on the birational class of X and the Grothendieck group of the semigroup of subspaces of rational functions on X. The constructed isomorphism moreover preserves the intersection numbers. This provides an alternative approach to b-divisors and their intersection theory.
A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by studying the complete intersection cone on a family of blow-ups of complex projective space, including the moduli space of stable six-pointed rational curves, $\bar{M}_{0,6}$, and the permutohedral or Losev-Manin moduli space of four-pointed rational curves. Our main result is that the movable and complete intersection cones coincide for the toric members of th…
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.
We study the following question: fix a sufficient general curve D of degree d in P^2, what is the least number of intersections between D and an irreducible curve of degree m? G. Xu proved this number i(d, m) is at least d - 2 for all m. This problem can be regarded as the algebraic part of Kobayashi conjecture on the hyperbolicity of P^2 D. We first improved Xu's bound with m fixed and then generalized his result to rational ruled surfaces.
We consider the stable ruled surface $S_1$ over an elliptic curve. There is a unique foliation on $S_1$ transverse to the fibration. The minimal self-intersection sections also define a 2-web. We prove that the 4-web defined by the fibration, the foliation and the 2-web is locally parallelizable.
We provide a general method for computing rational Chow rings of moduli of smooth complete intersections. We specialize this result in different ways: to compute the integral Picard group of the associated stack ; to obtain an explicit presentation of rational Chow rings of moduli of smooth complete intersections of codimension two; to prove old and new results on moduli of smooth curves of genus $\leq 5$ and polarized K3 surfaces of degree $\leq 8$.
We prove that (not necessarily rigid) Calabi-Yau threefolds defined over the rationals which contain sufficiently many elliptic ruled surfcaes are modular (under mild restrictions on the primes of bad reduction). Our proof uses the results of Dieulefait and Manoharmayum who proved modularity of rigid Calabi-Yau threefolds.
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.
We study the geometry of the Kontsevich compactification of stable maps to the Grassmannian of lines in the projective space. We consider a stratification of this space. As an application we compute the degree of the variety parametrizing rational ruled surfaces with a minimal directix of degree d/2-1 by intersecting divisors in the moduli space of stable maps. For example, there are 128054031872040 rational ruled sextics passing through 25 points in $\mathbb{P}^{3}$ with a minimal directrix of degree 2.
We prove that a complete intersection of $c$ very general hypersurfaces of degree at least two in $N$-dimensional complex projective space is not ruled (and therefore not rational) provided that the sum of the degrees of the hypersurfaces is at least $\tfrac{2}{3} N + c + 1$. To this end we consider a degeneration to positive characteristic, following Kollár. Our argument does not require a resolution of the singularities of the special fiber of the degeneration. It relies on a generalization of Kollár's "algebraic Morse lemma" that controls the dimensions of the second-order Thom-Boardman sin…
The family of varieties of minimal rational tangents on a quasi-homogeneous projective manifold is isotrivial. Conversely, are projective manifolds with isotrivial varieties of minimal rational tangents quasi-homogenous? We will show that this is not true in general, even when the projective manifold has Picard number 1. In fact, an isotrivial family of varieties of minimal rational tangents needs not be locally flat in differential geometric sense. This leads to the question for which projective variety Z, the Z-isotriviality of varieties of minimal rational tangents implies local flatness. O…
We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincare polynomials of the moduli spaces to Donaldson-Thomas invariants of the surface. In support of this result, we compute explicitly intersection Poincare polynomials for sheaves with rank two and three on ruled surfaces.
We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We introduce an algorithm for constructing examples where the upper bound is tight. As an application of our methods we improve an inequality on lattice polygons.
Let $X$ a complex projective variety of complex dimension $n$ with only isolated singularities of simply connected links. We show that we can endow the rational cohomology of the family of the $\overline{p}$-perverse intersection spaces $\{ I^{\overline{p}}X \}$ with compatible mixed Hodge structures.
In this short note we prove that in many cases the failure of a variety to be separably rationally connected is caused by the instability of the tangent sheaf (if there are no other obvious reasons). A simple application of the results proves that a smooth Fano complete intersection is separably rationally connected if and only if it is separably uniruled. In particular, a general such Fano complete intersection is separably rationally connected.
