Luis Fuentes Garcia, Manuel Pedreira Perez
We study the projective normality of a linearly normal special scroll R of degree d and speciality i over a smooth curve of genus g. If d>=4g-2i+1, we prove that the projective normality of the scroll is equivalent to the projective normality of its directrix curve of minimum degree.
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We study the projective normality of a linearly normal special scroll R of degree d and speciality i over a smooth curve X of genus g. We relate it with the Clifford index of the base curve X. If d>=4g-2i-Cliff(X)+1, i>=3 and R is smooth, we prove that the projective normality of the scroll is equivalent to the projective normality of its directrix curve of minimum degree.
We prove that the irregularity of a smooth abelian scroll whose dimension is at least half of that of the surrounding projective space is bounded by 2. In addition we discuss some existence results and open problems.
We study the projective models of complex K3 surfaces polarized by a line bundle L such that all smooth curves in |L| have non-general Clifford index. Such models are in a natural way contained in rational normal scrolls. We use this study to classify and describe all projective models of K3 surfaces of genus at most 10.
Gaussian scrolls, Gaussian flags and duality
本文研究高斯映射具有正维纤维的射影簇所对应的高斯涡卷与高斯旗，引入并探讨其类中的对偶性，揭示高斯涡卷与导出涡卷、切可展涡卷及反导出涡卷之间的对偶关系。
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.
The classification of projective elliptic line scrolls with the description of their singular loci is given. In particular we recover Atiyah Theorem by using classical methods.
On K3 fibred Calabi-Yau threefolds in weighted scrolls
本文分类了嵌入加权涡卷的低度加权 K3 曲面纤维化温和奇异 Calabi-Yau 三流形，推广 Mullet 结果，并研究射影退化与嵌入问题。
Let $X \subseteq \mathbb{P}^r$ be a scroll of codimension $e$ and degree $d$ over a smooth projective curve of genus $g$. The purpose of this paper is to prove a linear Castelnuovo-Mumford regularity bound that reg$(X) \leq d-e+1+g(e-1)$. This bound works over an algebraically closed field of arbitrary characteristic.
We give some results on quadratic normality of reducible curves canonically embedded and partially extend this study to their projective normality.
A well-known conjecture asserts that smooth threefolds $X\subset\{\mathbb P}^5$ are quadratically normal with the only exception of the Palatini scroll. As a corollary of a more general statement we obtain the following result, which is related to the previous conjecture: If $X\subset\{\mathbb P}^5$ is not quadratically normal, then its triple curve is reducible. Similar results are also given for higher dimensional varieties.
Let $C$ be an integral and projective curve whose canonical model $C'$ lies on a rational normal scroll $S$ of dimension $n$. We mainly study some properties on $C$, such as gonality and the kind of singularities, in the case where $n=2$ and $C$ is non-Gorenstein, and in the case where $n=3$, the scroll $S$ is smooth, and $C'$ is a local complete intersection inside $S$. We also prove that a rational monomial curve with just one singular point lies on a surface scroll if and only if its gonality is at most $3$, and that it lies on a threefold scroll if and only if its gonality is at most $4$.
We reduce the problem of the projective normality of polarized abelian varieties to check the rank of very explicit matrices. This allow us to prove some results on normal generation of primitive line bundles on abelian threefolds and fourfolds. We also give two situations where the projective normality always fails. Finally we make some conjecture.
Let $E$ be a vector bundle over a smooth curve $C$, and $S = \mathbb{P} E$ the associated projective bundle. We describe the inflectional loci of certain projective models $ψ\colon S \dashrightarrow \mathbb{P}^n$ in terms of Quot schemes of $E$. This gives a geometric characterisation of the Segre invariant $s_1 (E)$, which leads to new geometric criteria for semistability and cohomological stability of bundles over $C$. We also use these ideas to show that for general enough $S$ and $ψ$, the inflectional loci are all of the expected dimension. An auxiliary result, valid for a general subvarie…
We show that an abelian surface embedded in P^N by a very ample line bundle L of type (1,2d) is projectively normal if and only if d>=4. This completes the study of the projective normality of abelian surfaces embedded by complete linear systems.
In this paper we study examples of P^r-scrolls defined over primitively polarized K3 surfaces S of genus g, which arise from Brill-Noether theory of the general curve in the primitive linear system on S and from classical Lazarsfeld's results in. We show that such scrolls form an open dense subset of a component H of their Hilbert scheme; moreover, we study some properties of H (e.g. smoothness, dimensional computation, etc.) just in terms of the moduli space of such K3's and of the moduli space of semistable torsion-free sheaves of a given Mukai-vector on S. One of the motivation of this anal…
For a smooth projective variety $X\subset ¶^r$ embedded by the complete linear system, Property $N_p$ has been studied for a long time. On the other hand, Castelnuovo-Mumford regularity conjecture and related problems have been focused for a projective variety which is not necessarily linearly normal. This paper aims to explain the influence of Property $N_p$ on higher normality and defining equations of a smooth variety embedded by sub-linear system. Also we prove a claim about Property $N_p$ of surface scrolls which is a generalization of Green's work in \cite{Green} about Property $N_p$ for…
In this paper we study the projective normality of certain Artin-Schreier curves $Y_f$ defined over a field $\F$ of characteristic $p$ by the equations $y^q+y=f(x)$, $q$ being a power of $p$ and $f\in \F[x]$ being a polynomial in $x$ of degree $m$, with $(m,p)=1$. Many $Y_f$ curves are singular and so, to be precise, here we study the projective normality of appropriate projective models of their normalizations.
Hilbert schemes of suitable smooth, projective manifolds of low degree which are 3-fold scrolls over the Hirzebruch surface F_1 are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension and the general point of such a component is described.
In order to find useful information to complete the classification of Enriques-Fano threefolds, we will computationally study the singularities of some known Enriques-Fano threefolds of genus 6, 7, 8, 9, 10, 13 and 17. We will also deduce the projective normality of these threefolds.
We show that ample line bundles L on a g-dimensional simple abelian variety, with h^0(L)> 2^g.g!, give projectively normal embeddings, for all g>0.
In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.
We investigate the projective normality of smooth, linearly normal surfaces of degree 9. All non projectively normal surfaces which are not scrolls over a curve are classified. Results on the projective normality of surface scrolls are also given. One of the reasons that brought us to look at this question is our desire to find examples for a long standing problem in adjunction theory. Andreatta followed by a generalization by Ein and Lazarsfeld posed the problem of classifying smooth n-dimensional varieties (X,L) polarized with a very ample line bundle L, such that the adjoint linear system |…
One of the simplest examples of a smooth, non degenerate surface in P^4 is the quintic elliptic scroll. It can be constructed from an elliptic normal curve E by joining every point on E with the translation of this point by a non-zero 2-torsion point. The same construction can be applied when E is replaced by a (lineaerly normally embedded) abelian variety A. In this paper we ask the question when the resulting scroll Y is smooth. If A is an abelian surface embedded by a line bundle L of type (d_1,d_2) and r=d_1d_2, then we prove that for general A the scroll Y is smooth if r is at least 7 wit…
We prove some results on effective very ampleness and projective normality for some varieties with trivial canonical bundle. In the first part we prove an effective projective normality result for an ample line bundle on regular smooth four-folds with trivial canonical bundle. More precisely we show that for a regular smooth fourfold with trivial canonical bundle, $A^{\otimes 15}$ is projectively normal for $A$ ample. In the second part we emphasize on the projective normality of multiples of ample and globally generated line bundles on certain classes of known examples (upto deformation) of p…
Hilbert schemes of suitable smooth, projective 3-fold scrolls over the Hirzebruch surface F_e, with e > 1, are studied. An irreducible component of the Hilbert scheme parametrizing such varieties is shown to be generically smooth of the expected dimension and the general point of such a component is described. This article generalizes the study of Hilbert schemes done in arXiv:1110.5464 for e=1.
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 t…
On the projective normality of Ulrich bundles on some low-dimensional varieties
本文研究了 Ulrich 向量丛通过其 tautological 线丛的完全线性系嵌入时的射影丛的射影正规性，重点关注定义于曲线、$q=p_g=0$ 曲面及 $2$ 和 $3$ 维超曲面上的 Ulrich 丛。
In this article, we show that very ample line bundles of type (1,1,d) on an abelian threefold A, with NS(A)\simeq \ze, give projectively normal embeddings, if d=22,26 or when d\geq 28.
We classify the Ulrich vector bundles of arbitrary rank on smooth projective varieties of minimal degree. In the process, we prove the stability of the sheaves of relative differentials on rational scrolls.
We give necessary and sufficient criteria for a smooth Enriques surface S in P^r to be scheme-theoretically an intersection of quadrics. Moreover we prove in many cases that, when S contains plane cubic curves, the intersection of the quadrics containing S is the union of S and the 2-planes spanned by the plane cubic curves. We also give a new (very quick) proof of the projective normality of S if the degree of S is at least 12.
We study the projective normality of a minimal surface $X$ which is a ramified double covering over a rational surface $S$ with $\dim|-K_S|\ge 1$. In particular Horikawa surfaces, the minimal surfaces of general type with $K^2_X=2p_g(X)-4$, are of this type, up to resolution of singularities. Let $π$ be the covering map from $X$ to $S$. We show that the $\mathbb{Z}_2$-invariant adjoint divisors $K_X+rπ^*A$ are normally generated, where the integer $r\ge 3$ and $A$ is an ample divisor on $S$.
For a polarized abelian variety $(X,L)$, Z. Jiang and G. Pareschi introduce an invariant $β(X,L)$, called the basepoint-freeness threshold. Using this invariant, we show that a general polarized abelian variety $(X,L)$ of dimension $g$ is projectively normal if $χ(L) \geq 2^{2g-1}$ and the type of $L$ is not $(2,4,\dots,4)$. This bound is sharp since it is known that any polarized abelian variety of type $(2,4,\dots,4)$ is not projectively normal. We also give an application of $β(X,L)$ to the Infinitesimal Torelli Theorem for $Y \in |L|$.
Recall that a smooth complex projective curve has a very ample canonical bundle when it is non-hyperelliptic, and according to a theorem of M. Noether the resulting embedding is projectively normal. A theorem of Petri further asserts that the homogeneous ideal is generated by quadrics if the curve is neither trigonal nor a smooth plane quintic. In this note, we prove an analogue of Noether's theorem for the symmetric square of the curve - namely, the canonical bundle of the symmetric square determines a projectively normal embedding exactly when the curve itself is neither hyperelliptic, trigo…
Let $L$ be a very ample line bundle on a smooth curve $C$ of genus $g$ with $\frac{3g+3}{2}<°L\le 2g-5$. Then $L$ is normally generated if $°L>\max\{2g+2-4h^1(C,L), 2g-\frac{g-1}{6}-2h^1(C,L)\}$. Let $C$ be a triple covering of genus $p$ curve $C'$ with $C\stackrelϕ\to C'$ and $D$ a divisor on $C'$ with $4p<°D< \frac{g-1}{6}-2p$. Then $K_C(-ϕ^*D)$ becomes a very ample line bundle which is normally generated. As an application, we characterize some smooth projective surfaces.
On Normality of Projective Hypersurfaces with an Additive Action
本文研究具有诱导加法作用的射影超曲面，给出其正规性判别准则，并证明任意射影超曲面可作为某类超曲面开轨道补集的射影锥，同时引入基于杨图的构造方法。
Effective results on projective normality of the first and second secant varieties
本文针对光滑复射影簇由充分正线丛的完全线性系统嵌入时，第一和第二割线簇的射影正规性给出有效条件，并给出第一割线簇定义理想由三次式及3×3子式生成的有效条件。
The aim of this paper is two--fold. We first strongly improve our previous main result Theorem 3.1 in Arxiv 1702.00918v3 12Feb2018 ("Brill-Noether loci of rank two vector bundles on a general $ν$-gonal curve"), concerning classification of irreducible components of the Brill--Noether locus parametrizing rank 2 semistable vector bundles of suitable degrees $d$, with at least $d-2g+4$ independent global sections, on a general $ν$--gonal curve $C$ of genus $g$. We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which…
Ran proved that smooth codimension 2 varieties in ${\bf P}^{m+2}$ are $j$-normal if $(j+1)(3j-1)\le m-1$, in this paper we extend this result to small codimension projective varieties. Let $X$ be a r codimension subvariety of $\pro$, we prove that if the set $Σ_{(j+1)}$ of $(j+1)$-secants to $X$ through a generic external point is not empty, $2(r+1)j\leq m-r$ and $(j+1)((r+1)j-1)\leq m-1$ then $X$ is $j$-normal. If $X$ is given by the zero locus of a section of a rank $r$ vector bundle $E$ on $\pro$, we prove that $\textrm{deg} Σ_{j+1}=\frac{1}{(j+1)!}\prod_{i=0}^{j}c_{r}(E(-i))$. Moreover we …
We study projective surfaces $X \subset \mathbb{P}^r$ (with $r \geq 5$) of maximal sectional regularity and degree $d > r$, hence surfaces for which the Castelnuovo-Mumford regularity $\reg(C)$ of a general hyperplane section curve $C = X \cap \mathbb{P}^{r-1}$ takes the maximally possible value $d-r+3$. We show that each of these surfaces is either a cone over a curve $C \subset \mathbb{P}^{r-1}$ of maximal regularity or else a birational outer linear projection of a smooth rational surface scroll $\widetilde{X} \subset \mathbb{P}^{d+1}$. We prove that the Castelnuovo-Mumford regularity of th…
Triple solids and scrolls
研究光滑射影簇在度3有限态射下的卷积结构，证明维度限制并分类m=1和m=2情形，发现多种情况简化为Segre积。
In this note, we prove that for any finite dimensional vector space $V$ over $\mathbb {C}$, and for a finite cyclic group $G$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$ by a method using toric variety, and deduce the EGZ theorem as a consequence.
In this note, we prove that for any finite dimensional vector space $V$ over an algebraically closed field $k$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the $|G|$ is a unit in $k$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$.
If $G$ is a complex simply connected semisimple algebraic group and if $λ$ is a dominant weight, we consider the compactification $X_λ$ in the projectivisation of $\End(V(λ))$ obtained as the closure of the $G\times G$-orbit of the identity and we give necessary and sufficient conditions on the support of $λ$ so that $X_λ$ is normal; as well, we give necessary and sufficient conditions on the support of $λ$ so that $X_λ$ is smooth.
We show that if an ample line bundle L on a nonsingular toric 3-fold satisfies h^0(L+2K)=0, then L is normally generated. As an application, we show that the anti-canonical divisor on a nonsingular toric Fano 4-fold is normally generated.
A $h$-instanton sheaf on a closed subscheme $X$ of some projective space endowed with an ample and globally generated line bundle $\mathcal{O}_X(h)$ is a coherent sheaf whose cohomology table has a certain prescribed shape. In this paper we deal with $h$-instanton sheaves relating them to Ulrich sheaves. Moreover, we study $h$-instanton sheaves on smooth curves and surfaces, cyclic $n$-folds, Fano $3$-folds and scrolls over arbitrary smooth curves. We also deal with a family of monads associated to $h$-instanton bundles on varieties satisfying some mild extra technical conditions.
A Cremona transformation is a birational self-map of the projective space $ \mathbb{P}^{n} $. Cremona transformations of $ \mathbb{P}^{n} $ form a group and this group has a rational action on subvarieties of $ \mathbb{P}^{n} $ and hence on its Hilbert scheme. We study this action on the family of rational curves of $ \mathbb{P}^{3} $ and we prove the rectifiability of any one dimensional family. This shows that any uniruled surface is Cremona equivalent to a scroll and it answers a question of Bogomolov-Böhning related to the study of uniformly rational varieties. We provide examples of infin…
The Hilbert scheme of projective 3-folds of codimension 3 or more that are linear scrolls over the projective plane or over a smooth quadric surface or that are quadric or cubic fibrations over the projective line is studied. All known such threefolds of degree from 7 to 11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed. A relationship with the locus of good determinantal subschemes is investigated
We show that a general $n$-dimensional polarized abelian variety $(A,L)$ of a given polarization type and satisfying $ h^0(A, L) \geq \dfrac{8^n}{2} \cdot \dfrac{n^n}{n !}$ is projectively normal. In the process, we also obtain a sharp lower bound for the volume of a purely one-dimensional complex analytic subvariety in a geodesic tubular neighborhood of a subtorus of a compact complex torus.
Let $S$ be a rational surface with $\dim|-K_S|\ge 1$ and let $π: X\rightarrow S$ be a ramified cyclic covering from a nonruled smooth surface $X$. We show that for any integer $k\ge 3$ and ample divisor $A$ on $S$, the adjoint divisor $K_X+kπ^*A$ is very ample and normally generated. Similar result holds for minimal (possibly singular) coverings.
In this note, we prove that for the standard representation $V$of the Weyl group $W$ of a semi-simple algebraic group of type $A_n, B_n, C_n, D_n, F_4$ and $G_2$ over $\mathbb C$, the projective variety $\mathbb P(V^m)/W$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |W|}$, where $V^m$ denote the direct sum of $m$ copies of $V$. We also prove that for any finite group $G$ and for any finite dimentional representation $V$ over $\mathbb C$, the projective variety $P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes n!}$ as a conse…
In this paper, we show that the secant variety to a smooth projective variety embedded by a sufficiently positive line bundle is normal. As an application, we deduce that the secant variety to a general canonical curve of genus at least 7 is normal.
A long standing conjecture, known to us as the Eisenbud Goto conjecture, states that an n-dimensional variety embedded with degree $d$ in the $N$- dimensional projective space is $(d-(N-n)+1)$-regular in the sense of Castelnuovo-Mumford. In this work the conjecture is proved for all smooth varieties $X$ embedded by the complete linear system associated with a very ample line bundle $L$ such that $Δ(X,L) \le 5$ where $Δ(X,L) = \dim{X} + °{X} -h^0(L).$ As a by-product of the proof of the above result the projective normality of a class of surfaces of degree nine in $\Pin{5}$ which was left as an…
We provide two examples of smooth projective surfaces of tame CM type, by showing that any parameter space of isomorphism classes of indecomposable ACM bundles with fixed rank and determinant on a rational quartic scroll in projective 5-space is either a single point or a projective line. For surfaces of minimal degree and wild CM type, we classify rigid Ulrich bundles as Fibonacci extensions. For the rational normal scrolls S(2,3) and S(3,3), a complete classification of rigid ACM bundles is given in terms of the action of the braid group in three strands.
For a smooth curve of genus $g$ embedded by a line bundle of degree at least $2g+3$ we show that the ideal sheaf of the secant variety is 5-regular. This bound is sharp with respect to both the degree of the embedding and the bound on the regularity. Further, we show that the secant variety is projectively normal for the generic embedding of degree at least $2g+3$. We also give a conjectural description of the resolutions of the ideals of higher secant varieties.
Let $(X, A)$ be a nonsingular polarized toric 3-fold. We show that if the adjoint bundle of $A$ has no glabal section, then all ample line bundles on $X$ are normally generated. Even if the adjoint bundle is effective, if it is not big, then it is shown the normal generation. In particular, we show that all ample line bundles on a nonsingular toric Fano 3-fold are normally generated.
We prove that in characteristic zero the multiplication of sections of dominant line bundles on a complete symmetric variety $X=\bar{G/H}$ is a surjective map. As a consequence the cone defined by a complete linear system over $X$, or over a closed $G$ stable subvariety of $X$ is normal. This gives an affirmative answer to a question raised by Faltings. A crucial point of the proof is a combinatorial property of root systems.
We develop in this article an algorithm that, given a projective curve $C$, computes a \textit{gonal map}, that is, a finite morphism from $C$ to the projective line of minimal degree. Our method is based on the computation of scrollar syzygies of canonical curves. We develop an improved version of our algorithm for curves with a unique gonal map and we discuss a characterization of such curves in terms of Betti numbers. Finally, we derive an efficient algorithm for radical parametrization of curves of gonality $\le 4$.
We prove that the multiplication of sections of globally generated line bundles on a model wonderful variety M of simply connected type is always surjective. This follows by a general argument which works for every wonderful variety and reduces the study of the surjectivity for every couple of globally generated line bundles to a finite number of cases. As a consequence, the cone defined by a complete linear system over M or over a closed G-stable subvariety of M is normal. We apply these results to the study of the normality of the compactifications of model varieties in simple projective spa…
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 study graded Betti numbers of any nondegenerate 3-regular algebraic set $X$ in a projective space $\mathbb P^{n}$. More concretely, via Generic initial ideals (Gins) method we mainly consider `tailing' Betti numbers, whose homological index is not less than $\mathrm{codim}(X,\mathbb P^{n})$. For this purpose, we first introduce a key definition `$\mathrm{ND(1)}$ property', which provides a suitable ground where one can generalize the concepts such as `being nondegenerate' or `of minimal degree' from the case of varieties to the case of more general closed subschemes and give a…
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