Rational Normal Scrolls Research Articles

Castelnuovo–Mumford regularity and splitting criteria for logarithmic bundles over rational normal scroll surfaces

We introduce and study a notion of Castelnuovo–Mumford regularity suitable for rational normal scroll surfaces. In this setting we prove analogs of some classical properties. We prove splitting criteria for coherent sheaves and a characterization of Ulrich bundles. Finally we study logarithmic bundles associated to arrangements of lines and rational curves.

On codimension one holomorphic distributions on compact toric orbifolds

The number of singularities, counted with multiplicity, of a generic codimension one holomorphic distribution on a compact toric orbifold is determined. As a consequence, we provide a classification for regular distributions on rational normal scrolls and weighted projective spaces. Additionally, under specific conditions, we prove that the singular set of a codimension one holomorphic foliation on a compact toric orbifold admits at least one irreducible component of codimension two, and we also present a Darboux–Jouanolou type integrability theorem for codimension one holomorphic foliations. Our results are exemplified through various illustrative examples.

Characterization of projective varieties beyond varieties of minimal degree and del Pezzo varieties

Varieties of minimal degree and del Pezzo varieties are basic objects in projective algebraic geometry. Those varieties have been characterized and classified for a long time in many aspects. Motivated by the question “which varieties are the most basic and simplest except the above two kinds of varieties in view of geometry and syzygies?”, we give an upper bound of the graded Betti numbers in the quadratic strand and characterize the extremal cases.The extremal varieties of dimension n, codimension e, and degree d are exactly characterized by the following two types:(i)Varieties with d=e+2, depthX=n, and Green-Lazarsfeld index a(X)=0,(ii)Arithmetically Cohen-Macaulay varieties with d=e+3. This is a generalization of G. Castelnuovo, G. Fano, and E. Park's results on the number of quadrics and an extension of the characterizations of varieties of minimal degree and del Pezzo varieties in view of linear syzygies of quadrics due to K. Han and S. Kwak ([6,8,30,16]).In addition, we show that every variety X that belongs to (i) or (ii) is always contained in a unique rational normal scroll Y as a divisor. Also, we describe the divisor class of X in Y.

On the genus of projective curves not contained in hypersurfaces of given degree

Fix integers rge 4 and ige 2 (for r=4 assume ige 3). Assume that the rational number s defined by the equation left( {begin{array}{c}i+1 2end{array}}right) s+(i+1)=left( {begin{array}{c}r+i iend{array}}right) is an integer. Fix an integer dge s. Divide d-1=ms+epsilon, 0le epsilon le s-1, and set G(r;d,i):=left( {begin{array}{c}m 2end{array}}right) s+mepsilon. As a number, G(r; d, i) is nothing but the Castelnuovo’s bound G(s+1;d) for a curve of degree d in {mathbb {P}}^{s+1}. In the present paper we prove that G(r; d, i) is also an upper bound for the genus of a reduced and irreducible complex projective curve in {mathbb {P}}^r, of degree dgg max {r,i}, not contained in hypersurfaces of degree le i. We prove that the bound G(r; d, i) is sharp if and only if there exists an integral surface Ssubset {mathbb {P}}^r of degree s, not contained in hypersurfaces of degree le i. Such a surface, if existing, is necessarily the isomorphic projection of a rational normal scroll surface of degree s in {mathbb {P}}^{s+1}. The existence of such a surface S is known for rge 5, and 2le i le 3. It follows that, when rge 5, and i=2 or i=3, the bound G(r; d, i) is sharp, and the extremal curves are isomorphic projection in {mathbb {P}}^r of Castelnuovo’s curves of degree d in {mathbb {P}}^{s+1}. We do not know whether the bound G(r; d, i) is sharp for i>3.

On the normal sheaf of Gorenstein curves

We show that any tetragonal Gorenstein integral curve is a complete intersection in its respective 3-fold rational normal scroll S, implying that the normal sheaf on C embedded in S, and in Pg−1 as well, is unstable for g≥5, provided that S is smooth. We also compute the degree of the normal sheaf of any singular reduced curve in terms of the Tjurina and Deligne numbers, providing a semicontinuity of the degree of the normal sheaf over suitable deformations, revisiting classical results of the local theory of analytic germs.

Fiber cones of rational normal scrolls are Cohen–Macaulay

In this paper, we show that the fiber cones of rational normal scrolls are Cohen–Macaulay. As an application, we compute their Castelnuovo–Mumford regularities and $$\varvec{a}$$ -invariants, as well as the reduction number of the defining ideals of the rational normal scrolls. We also characterize the Gorensteinness of the fiber cone.

Space curves on surfaces with ordinary singularities

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.

H-instanton bundles on three-dimensional polarized projective varieties

We propose a notion of instanton bundle (called H-instanton bundle) on any projective variety of dimension three polarized by a very ample divisor H, that naturally generalizes the ones on P3 and on the flag threefold F(0,1,2). We briefly discuss the cases of Veronese and Fano threefolds. Then we deal with H-instanton bundles E on three-dimensional rational normal scrolls S(a0,a1,a2). We give a monadic description of H-instanton bundles and we prove the existence of μ-stable H-instanton bundles on S(a0,a1,a2) for any admissible charge k=c2(E)H. Then we deal in more detail with S(a,a,b) and S(a0,a1,a2) with a0+a1>a2 and even degree. Finally we describe a nice component of the moduli space of μ-stable bundles whose points represent H-instantons.

Blow-up algebras of secant varieties of rational normal scrolls

In this paper, we are mainly concerned with the blow-up algebras of the secant varieties of balanced rational normal scrolls. In the first part, we give implicit defining equations of their associated Rees algebras and fiber cones. Consequently, we can tell that the fiber cones are Cohen–Macaulay normal domains. Meanwhile, these fiber cones have rational singularities in characteristic zero, and are F-rational in positive characteristic. The Gorensteinness of the fiber cones can also be characterized. In the second part, we compute the Castelnuovo–Mumford regularities and $${\varvec{a}}$$ -invariants of the fiber cones. We also present the reduction numbers of the ideals defined by the secant varieties.

The relative canonical resolution: Macaulay2-package, experiments and conjectures

This short note provides a quick introduction to relative canonical resolutions of curves on rational normal scrolls. We present our Macaulay2-package which computes the relative canonical resolution associated to a curve and a pencil of divisors. Most of our experimental data can be found on a dedicated webpage. We end with a list of conjectural shapes of relative canonical resolutions. In particular, for curves of genus $g=n\cdot k +1$ and pencils of degree $k$ for $n\ge 1$, we conjecture that the syzygy divisors on the Hurwitz space $\mathscr{H}_{g,k}$ constructed by Deopurkar and Patel all have the same support.

Rank 3 quadratic generators of Veronese embeddings

Let $L$ be a very ample line bundle on a projective scheme $X$ defined over an algebraically closed field $\Bbbk$ with ${\rm char}\,\Bbbk \neq 2$. We say that $(X,L)$ satisfies property $\mathsf {QR}(k)$ if the homogeneous ideal of the linearly normal embedding $X \subset {\mathbb {P}} H^{0} (X,L)$ can be generated by quadrics of rank less than or equal to $k$. Many classical varieties, such as Segre–Veronese embeddings, rational normal scrolls and curves of high degree, satisfy property $\mathsf {QR}(4)$. In this paper, we first prove that if ${\rm char}\,\Bbbk \neq 3$ then $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (d))$ satisfies property $\mathsf {QR}(3)$ for all $n \geqslant 1$ and $d \geqslant 2$. We also investigate the asymptotic behavior of property $\mathsf {QR}(3)$ for any projective scheme. Specifically, we prove that (i) if $X \subset {\mathbb {P}} H^{0} (X,L)$ is $m$-regular then $(X,L^{d} )$ satisfies property $\mathsf {QR}(3)$ for all $d \geqslant m$, and (ii) if $A$ is an ample line bundle on $X$ then $(X,A^{d} )$ satisfies property $\mathsf {QR}(3)$ for all sufficiently large even numbers $d$. These results provide affirmative evidence for the expectation that property $\mathsf {QR}(3)$ holds for all sufficiently ample line bundles on $X$, as in the cases of Green and Lazarsfeld's condition $\mathrm {N}_p$ and the Eisenbud–Koh–Stillman determininantal presentation in Eisenbud et al. [Determinantal equations for curves of high degree, Amer. J. Math. 110 (1988), 513–539]. Finally, when ${\rm char}\,\Bbbk = 3$ we prove that $({\mathbb {P}}^{n} , \mathcal {O}_{{\mathbb {P}}^{n}} (2))$ fails to satisfy property $\mathsf {QR}(3)$ for all $n \geqslant 3$.

Reconstruction of rational ruled surfaces from their silhouettes

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 P3 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 P3 of these surfaces by exploiting the information contained in the singularities of the apparent contour.

Blowup algebras of rational normal scrolls

We determine the equations of the blowup of P n \mathbb {P}^{n} along a d d -fold rational normal scroll S \mathcal {S} , and we prove that the Rees ring and special fiber ring of S ⊆ P n \mathcal {S}\subseteq \mathbb {P}^{n} are Koszul algebras.

The Rees algebra of parametric curves via liftings

We study the defining equations of the Rees algebra of ideals arising from curve parametrizations in the plane and in rational normal scrolls, inspired by the work of Madsen and Kustin, Polini and Ulrich. The curves are related by work of Bernardi, Gimigliano, and Idá, and we use this framework to relate the defining equations.

Scrolls containing binary curves

We study families of scrolls containing a given rational curve and families of rational curves contained in a fixed scroll via a stratification in terms of the degree of the induced map onto $$\mathbb P^1$$ and we prove that there is no rational normal scroll of minimal degree and of dimension $$\le \frac{n}{2}$$ containing a general binary curve in $$\mathbb P^n$$.

Equations and Syzygies of K3 Carpets and Unions of Scrolls

We describe the equations and Grobner 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 Grobner bases and syzygies for these objects over the integers and this lets us treat them in all characteristics simultaneously.

Mixed multiplicities, Segre numbers and Segre classes

Based on classical results of Rees and on multivariate Hilbert polynomials, we define new mixed multiplicities of two arbitrary ideals in a local ring (A,m) and use them to express the local degrees of all varieties appearing in the Gaffney–Gassler construction of Segre cycles. We prove that the classical mixed multiplicities of m and an arbitrary ideal I, which are a special case of the new ones, are equal to the generalized Samuel multiplicities of an ideal in the Rees algebra RI(A). This equality is used to improve a result of Jeffries, Montaño and Varbaro on the degree of the fiber cone of an ideal.We conclude the paper with formulas (and their inverses) which express the degrees of Segre classes of subschemes of arbitrary projective varieties by generalized Samuel multiplicities or by classical mixed multiplicities. Using the mixed multiplicities of balanced rational normal scrolls, which have been computed by Hoang and Lam, we find the mixed multiplicities of all rational normal scrolls as well as their Segre classes and their generalized Samuel multiplicities.

On gonality, scrolls, and canonical models of non-Gorenstein curves

Let C be an integral and projective curve; and let $$C'$$ be its canonical model. We study the relation between the gonality of C and the dimension of a rational normal scroll S where $$C'$$ can lie on. We are mainly interested in the case where C is singular, or even non-Gorenstein, in which case $$C'\not \cong C$$ . We first analyze some properties of an inclusion $$C'\subset S$$ when it is induced by a pencil on C. Afterwards, in an opposite direction, we assume $$C'$$ lies on a certain scroll, and check some properties C may satisfy, such as gonality and the kind of its singularities. At the end, we prove that a rational monomial curve C has gonality d if and only if $$C'$$ lies on a $$(d-1)$$ -fold scroll.

Correspondence Scrolls

This paper initiates the study of a class of schemes that we call correspondence scrolls, which includes the rational normal scrolls and linearly embedded projective bundle of decomposable bundles, as well as degenerate K3 surfaces, Calabi-Yau 3-folds, and many other examples.

Some loci of rational cubic fourfolds

In this paper we investigate the divisor $${\mathcal {C}}_{14}$$ inside the moduli space of smooth cubic hypersurfaces in $${\mathbb {P}}^5$$ , whose general element is a smooth cubic containing a smooth quartic rational normal scroll. By showing that all degenerations of quartic scrolls in $${\mathbb {P}}^5$$ contained in a smooth cubic hypersurface are surfaces with one apparent double point, we prove that every cubic hypersurface contained in $${\mathcal {C}}_{14}$$ is rational. Combining our proof with the Hodge theoretic definition of $${\mathcal {C}}_{14}$$ , we deduce that on a smooth cubic fourfold every class $$T\in {\text {H}}^{2,2}(X,\mathbb {Z})$$ with $$T^2=10$$ and $$T\cdot h^2=4$$ is represented by a (possibly reducible) surface of degree four which has one apparent double point. As an application of our results and of the construction of some explicit examples, we also prove that the Pfaffian locus is not open in $${\mathcal {C}}_{14}$$ .