arXiv subjects

Hossein Movasati

Publications and source records attributed to Hossein Movasati.

At least 19 recordsLinked to original sources

arXiv2026

On a counterexample to a conjecture of J. Harris for octic surfaces

We take a sum $C_1+r C_2,\ r\in\Q$ of a line $C_1$ and a complete intersection curve $C_2$ of type $(3,3)$ inside the octic Fermat surface and with no intersection points. We gather strong evidences to the fact that for all except a finite number of $r$, the Noether-Lefschetz loci attached to the cohomology classes of $C_1+r C_2$ are set theoretically distinct $31$ codimensional subvarieties intersecting each other in a $32$ codimensional subvariety of the ambient space. The maximum codimension for components of the Noether-Lefschetz locus in this case is $35$, and hence, we provide a possible counterexample to a conjecture of J. Harris.

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arXiv2026

Hasse-Witt invariants of Calabi-Yau varieties

We define the Hasse-Witt invariant of Calabi-Yau varieties in two different ways. The first method is through Cartier operator and the second method is through the theory of Calabi-Yau modular forms developed by the third author. We conjecture that these two definitions are equivalent and provide many examples of Calabi-Yau varieties in support of this conjecture.

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arXiv2025

Leaf schemes and Hodge loci

This is a collection of articles, written as sections, on arithmetic properties of differential equations, holomorphic foliations, Gauss-Manin connections and Hodge loci. Each section is independent from the others and it has its own abstract and introduction and the reader might get an insight to the text by reading the introduction of each section. The main connection between them is through comments in footnotes. Our major aim is to develop a theory of leaf schemes over finitely generated subrings of complex numbers, such that the leaves are also equipped with a scheme structure. We also aim to formulate a local-global conjecture for leaf schemes.

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arXiv2025

On a Hodge locus

There are many instances such that deformation space of the homology class of an algebraic cycle as a Hodge cycle is larger than its deformation space as algebraic cycle. This phenomena can occur for algebraic cycles inside hypersurfaces, however, we are only able to gather evidences for it by computer experiments. In this article we describe one example of this for cubic hypersurfaces. The verification of the mentioned phenomena in this case is proposed as the first GADEPs problem. The main goal is either to verify the (variational) Hodge conjecture in such a case or gather evidences that it might produce a counterexample to the Hodge conjecture.

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arXiv2025

Grothendieck-Katz conjecture

In this article we prove that linear differential equations with only algebraic solutions have zero $m$-curvature modulo $p^k$ for all except a finite number of primes $p$ and all $k,m\in\mathbb N$ with ${\rm ord}_pm!\geq k$. This provides us with a reformulation of Grothendieck-Katz conjecture with stronger hypothesis.

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arXiv2025

Ramanujan vector field

In this article we prove that for all primes $p\not=2,3$, the Ramanujan vector field has an invariant algebraic curve and then we give a moduli space interpretation of this curve in terms of Cartier operator acting on the de Rham cohomology of elliptic curves. The main ingredients of our study are due to Serre, Swinnerton-Dyer and Katz in 1973. We aim to generalize this for the theory of Calabi-Yau modular forms, which includes the generating function of genus $g$ Gromov-Witten invariants. The integrality of $q$-expansions of such modular forms is still a main conjecture which has been only established for special Calabi-Yau varieties, for instance those whose periods are hypergeometric functions. For this the main tools are Dwork's theorem. We present an alternative project which aims to prove such integralities using modular vector fields and Gauss-Manin connection in positive characteristic.

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arXiv2025

Local-global principle for leaf schemes

We study Hodge loci as leaf schemes of foliations. The main ingredient is the Gauss-Manin connection matrix of families of projective varieties. We also aim to investigate a conjecture on the ring of definition of leaf schemes and its consequences such as the algebraicity of leaf schemes (Cattani-Deligne-Kaplan theorem in the case of Hodge loci). This conjecture is a consequence of a local-global principle for leaf schemes.

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arXiv2022

Computing the lines of a smooth cubic surface

We give an explicit formula for the $27$ lines of a smooth cubic surface near the Fermat surface. Our formula involves convergent power series with coefficients in the extension of rational numbers with the sixth root of unity. Our main tool is the Artinian Gorenstein ring of socle two attached to such lines.

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arXiv2022

Smooth points of the space of plane foliations with a center

We prove that a logarithmic foliation corresponding to a generic line arrangement of $d+1 \geq 3$ lines in the complex plane, with pairwise natural and co-prime residues, is a smooth point of the center set of plane foliations (vector fields) of degree $d$.

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arXiv2022

Periods of families of curves in threefolds

Clemens' conjecture states that the the number of rational curve in a generic quintic threefold is finite. If it is false we prove that certain periods of rational curves in such a quintic threefold must vanish. Our method is based on a generalization of a proof of Max Noether's theorem using infinitesimal variation of Hodge structures and its reformulation in terms of integrals and Gauss-Manin connection.

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arXiv2021

Gauss-Manin Connection in Disguise: Dwork Family

We study the moduli space $\textsf{T}$ of the Calabi-Yau $n$-folds arising from the Dwork family and enhanced with bases of the $n$-th de Rham cohomology with constant cup product and compatible with Hodge filtration. We also describe a unique vector field $\textsf{R}$ in $\textsf{T}$ which contracted with the Gauss-Manin connection gives an upper triangular matrix with some non-constant entries which are natural generalizations of Yukawa couplings. For $n=1,2$ we compute explicit expressions of $\textsf{R}$ and give a solution of $\textsf{R}$ in terms of quasi-modular forms. The moduli space $\textsf{T}$ is an affine variety and for $n=4$ we give explicit coordinate system for $\textsf{T}$ and compute the vector field $\textsf{R}$ and the $q$-expnasion of its solution.

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arXiv2021

Gauss-Manin connection in disguise: Quasi Jacobi forms of index zero

We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomolgy with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields.

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arXiv2021

Algebraic curves and foliations

Consider a field $k$ of characteristic $0$, not necessarily algebraically closed, and a fixed algebraic curve $f=0$ defined by a tame polynomial $f\in k[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic foliations in the plane ${mathbb A}^2_\k$ having $f=0$ as a fixed invariant curve is generated as $k[x,y]$-module by at most four elements, three of them are the trivial foliations $fdx,fdy$ and $df$. Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen-Suslin theorem, we show that for a suitable field extension $K$ of $k$ such a module over $K[x,y]$ is actually generated by two elements, and therefore, such curves are free divisors in the sense of K. Saito. After performing Groebner basis for this module, we observe that in many well-known examples, $K=k$.

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arXiv2020

On reconstructing subvarieties from their periods

We give a new practical method for computing subvarieties of projective hypersurfaces. By computing the periods of a given hypersurface X, we find algebraic cohomology cycles on X. On well picked algebraic cycles, we can then recover the equations of subvarieties of X that realize these cycles. In practice, a bulk of the computations involve transcendental numbers and have to be carried out with floating point numbers. However, if X is defined over algebraic numbers then the coefficients of the equations of subvarieties can be reconstructed as algebraic numbers. A symbolic computation then verifies the results. As an illustration of the method, we compute generators of the Picard groups of some quartic surfaces. A highlight of the method is that the Picard group computations are proved to be correct despite the fact that the Picard numbers of our examples are not extremal.

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arXiv2020

Gauss-Manin connection in disguise: Genus two curves

We describe an algebra of meromorphic functions on the Siegel domain of genus two which contains Siegel modular forms for an arithmetic index six subgroup of the symplectic group and it is closed under three canonical derivations of the Siegel domain. The main ingredients of our study are the moduli of enhanced genus two curves, Gauss-Manin connection and the modular vector fields living on such moduli spaces.

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arXiv2019

Convolution of Picard-Fuchs equations

We determine explicit generators for a cohomology group constructed from a solution of a fuchsian linear differential equation and describe its relation with cohomology groups with coefficients in a local system. In the parameterized case, this yields into an algorithm which computes new fuchsian differential equations from those depending on multi-parameters. This generalizes the classical convolution of solutions of fuchsian differential equations.

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arXiv2019

Periods of linear algebraic cycles

In this article we use a theorem of Carlson and Griffiths and compute periods of linear algebraic cycles inside the Fermat variety of even dimension $n$ and degree $d$. As an application, for examples of $n$ and $d$ we prove that the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space. We also check the same statement for hypersurfaces containing a complete intersection algebraic cycle. Our result confirms the Hodge conjecture for Hodge cycles obtained by the monodromy of the homology class of such algebraic cycles. This is known as the variational Hodge conjecture.

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arXiv2019

Special components of Noether-Lefschetz loci

We take a sum $C_1+r C_2,\ r\in\mathbb Q$ of a line $C_1$ and a complete intersection curve $C_2$ of type $(3,3)$ inside a smooth surface of degree $8$ and with $C_1\cap C_2=\emptyset$. We gather evidences to the fact that for all except a finite number of $r$, the Noether-Lefschetz loci attached to the cohomology classes of $C_1+ r C_2$ are distinct $31$ codimensional subvarieties intersecting each other in a $32$ codimensional subvariety of the ambient space. The maximum codimension for components of the Noether-Lefschetz locus in this case is $35$, and hence, we provide a conjectural description of a counterexample to a conjecture of J. Harris. The methods used in this paper also produce in a rigorous way an infinite number of general components passing through the point representing the Fermat surface of degree $\leq 9$, and many non-reduced components for such degrees.

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