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Ziv Ran

Publications and source records attributed to Ziv Ran.

At least 19 recordsLinked to original sources
arXiv2025

A rank-$2$ vector bundle on ${\mathbb P}^2\times {\mathbb P}^2$ and projective geometry of nonclassical Enriques surfaces in characteristic 2

We study zero-sets of a particular family rank-2 indecomposable vector bundles $E$ on $\mathbb P^2 \times\mathbb P ^2$ in characteristic 2, introduced in our earlier paper. We show that the zero-sets of a suitable twist of $E$ form a family of nonclassical smooth Enriques surfaces of bidegree $(4, 4)$ whose general member is ordinary in the sense that Frobenius acts isomorphically on $H^1$, and which admits a divisor consisting of smooth supersingular surfaces (Frobenius acts as zero). We show further that every nonclassical Enriques surface of bidegree $(4, 4)$ that is bilinearly normal arises as a zero-set in this way.

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arXiv2024

Gaussian scrolls, Gaussian flags and duality

A projective variety whose Gauss map has positive dimensional fibres corresponds to a special kind of scroll called \emph{Gaussian}. A Gaussian scroll is a member of a canonical derived \emph{ Gaussian flag}. We introduce a duality in the class of Gaussian scrolls and flags and study its consequences. In particular, a Gaussian scroll is dual to the derived or tangent developable scroll of a Gaussian scroll in the dual projective space, and is the 'leading edge' or antiderived scroll of its derived stationary scroll.

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arXiv2024

Regular and rigid curves on some Calabi-Yau and general-type complete intersections

Let $X$ be either a general hypersurface of degree $n+1$ in $\mathbb P^n$ or a general $(2,n)$ complete intersection in $\mathbb P^{n+1}, n\geq 4$. We construct balanced rational curves on $X$ of all high enough degrees. If $n=3$ or $g=1$, we construct rigid curves of genus $g$ on $X$ of all high enough degrees. As an application we construct some rigid bundles on Calabi-Yau threefolds. In addition, we construct some low-degree balanced rational curves on hypersurfaces of degree $n + 2$ in $\mathbb P^n$.

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arXiv2023

Polarized interpolation and normal postulation for curves on Fano hypersurfaces

A general hypersurface $X$ of degree $\leq n$ in projective space contains curves $C$ of any genus $g\geq 0$ and sufficiently large degree depnding on $g$ whose normal and conormal bundles have good postulation or natural cohomology in the sense that each twist has either $H^0=0$ or $H^1=0$. This implies a polarized version of the interpolation property for $C$ on $X$.

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arXiv2022

Curves with stable or semistable normal bundle on Fano hypersurfaces

For every $n\geq 3, g\geq 1$ and all large enough $e$ depending on $n,g$, there exist curves of genus $g$, degree $e$ in a general hypersurface of degree $n$ in $\mathbb P^n$, or in $\mathbb P^n$ itself, whose whose normal bundle $N$ is stable, as is any sufficiently general full-rank subsheaf of $N$. For $g=1$, $N$ is semi-stable. On general hypersurface of degree $d< n$ in $\mathbb P^n$, such that a certain arithmetical condition on $d,n, g $ holds, there exists an arithmetical progression of $e$ values so that curves of degree $e$ and genus $g$ with semistable normal bundle exist. Previous results were restricted to certain cases with ambient space $\P^n$

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arXiv2022

Bundles on caudate curves

We study vector bundles on curves with rational tails and their smoothings and give a sufficient condition for the general fibre to be balanced.

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arXiv2022

Interpolation of curves on Fano hypersurfaces

On a general hypersurface of degree $d\leq n$ in $\mathbb P^n$ or $\mathbb P^n$ itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number $t$ of general points or incident to a general collection of subvarieties of suitable codimensions. In some cases we also show that the family of curves through $t$ fixed points has general moduli as family of $t$-pointed curves. These results imply positivity of certain intersection numbers on Kontsevich spaces of stable maps. An arithmetical appendix by M. C. Chang descibes the set of numerical characters ($n, d$, curve degree, genus) to which our results apply.

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arXiv2022

Complexes, residues and obstructions for log-symplectic manifolds

We consider compact Kählerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic structure $Φ$, a generically nondegenerate closed 2-form with simple poles on a divisor $D$ with local normal crossings. A simple linear inequality involving the iterated Poincaré residues of $Φ$ at components of the double locus of $D$ ensures that the pair $(X, Φ)$ has unobstructed deformations and that $D$ deforms locally trivially.

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arXiv2022

Rigid and stably balanced curves on Calabi-Yau and general-type hypersurfaces

A curve $C$ on a variety $X$ is stably balanced if the slopes of the Harder-Narasimhan filtration of its normal bundle $N$ are contained in an interval of length 1. For each $d\geq n+1$ we construct some regular families of pairs $(C, X)$ of the expected dimension with $X$ a hypersurface of degree $d$ in $\mathbb P^n$ and $C$ a stably balanced rigid curve on $X$, such that the family of hypersurfaces $X$ is smooth codimension $h^1(N)$ in the space of hypersurfaces.

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arXiv2022

Differential complexes and Hodge theory on log-symplectic manifolds

We study certain complexes of differential forms, including reverse de Rham complexes, on (real or complex) Poisson manifolds, especially holomorphic log-symplectic ones. We relate these to the degeneracy divisor and rank loci of the Poisson bivector. In some good holomorphic cases we compute the local cohomology of these complexes. In the Kahlerian case, we deduce a relation between the multiplicity loci of the degeneracy divisor and the Hodge numbers of the manifold. We also show that vanishing of one of these Hodge numbers is related tounobstructed deformations of the normalized degeneracy divisor with its induced Poisson structure.

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arXiv2022

On the size and local equations of fibres of general projections

For a general birational projection of a smooth nondegenerate projective $n$-fold from $\mathbb P^{n+c}$ to $\mathbb P^m$, $n<m\leq(n+c)/2$, all fibres have total length asymptotically bounded by $2^{\sqrt{n}+1} $ and the fibres are locally defined by linear and quadratic equations.

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arXiv2021

Balanced rational curves and minimal rational connectedness of Fano hypersurfaces

On a general Fano hypersurface in projective space, we determine for infinitely many $k$ the minimal degree $e$ of a rational curve through a general collection of $k$ points. In the case of a hypersurface of index 1, our results hold for all $k\geq 1$. In an appendix, M.C. Chang proves an arithmetical result which implies that in the case of index $>1$, the density of the set of curve degrees $e$ covered by our method is approximately $\frac{(n-d)(d-\frac{5}{2})}{(n-2)d}$.

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arXiv2021

Interpolation of rational scrolls

We show in many cases that there exist rational scrolls which are balanced, i.e. they contain the expected number of general linear spaces as rulings. For example, there exist balanced scrolls of degree $mk+1$ and fibre dimension $k$ in $¶^{2k+1}$ for all $m\geq 1$.

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arXiv2020

A Bogomolov unobstructedness theorem for log-symplectic manifolds in general position

We consider compact Kählerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure $Π$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor $D(Π)$. We prove that $(X, Π)$ has unobsrtuced deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on $H^2$ of the open symplectic manifold $X\setminus D(Π)$, and in fact coincides with this $H^2$ provided the Hodge number $h^{2,0}_X=0$, and finally that the degeneracy locus $D(Π)$ deforms locally trivially under deformations of $(X, Π)$. It has been pointed out that the general position hypothesis in the original paper is not strong enough and this is corrected in an appended erratum/corrigendum to the revised version.

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arXiv2020

Incident rational curves

We study families of rational curves on an algebraic variety satisfying incidence conditions. We prove an analogue of bend-and-break: that is, we show that under suitable conditions, such a family must contain reducibles. In the case of curves in $¶^n$ incident to certain complete intersections, we prove the family is irreducible.

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arXiv2020

Low-degree Rational curves on hypersurfaces in projective spaces and their degenerations

We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.

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