Searcharxiv⌕ Search

arXiv subjects

Remke Kloosterman

Publications and source records attributed to Remke Kloosterman.

At least 19 recordsLinked to original sources
arXiv2026

The Ciliberto-Di Gennaro conjecture for $d=5$

The Ciliberto-Di Gennaro conjecture predicts that a nodal hypersurface of degree $d\geq 3$ with at most $2(d-2)(d-1)$ nodes is either factorial, or contains a plane and has at least $(d-1)^2$ nodes, or contains a quadric surface and has $2(d-2)(d-1)$ nodes. This conjecture is classically known for $d=3,4$. In 2022 the author proved this conjecture for $d\geq 7$ by the author. Kvitko announced a proof for $d=6$ in 2025. In this paper we prove the conjecture for the remaining open value of $d$, namely $d=5$.

math.AG↗
arXiv2026

Supersingular elliptic surfaces and Infinitesimal Torelli

In 1981 Katsura presented a classification of non-rational Jacobian elliptic surfaces which admit a base change which is rational. In 2004 we presented a classification of Jacobian regular elliptic surfaces which do not satisfy infinitesimal Torelli. These classifications of quite different properties turn out to be very similar. In this paper we use an argument exploiting the product-quotient structure of these examples to prove simultaneously that Katsura's examples are Artin supersingular, and to give a new proof that our examples do not satisfy infinitesimal Torelli.

math.AG↗
arXiv2025

Determining explicitly the Mordell-Weil group of certain rational elliptic surfaces

Let $A,B$ be nonzero rational numbers. Consider the elliptic curve $E_{A,B}/\mathbb{Q}(t)$ with Weierstrass equation $y^2=x^3+At^6+B$. An algorithm to determine $\mathrm{rank } E_{A,B}(\mathbb{Q}(t))$ as a function of $(A,B)$ was presented in a recent paper by Desjardins and Naskrecki. We will give a different and shorter proof for the correctness of that algorithm, using a more geometric approach and discuss for which classes of examples this approach might be useful.

math.NT↗
arXiv2024

Hodge loci associated with linear subspaces intersecting in codimension one

Let $X\subset \mathbb{P}^{2k+1}$ be a smooth hypersurface containing two k-dimensional linear spaces $\Pi_1,\Pi_2$ intersecting in codimension one. In this paper we study the question whether the Hodge loci $NL([\Pi_1]+\lambda[\Pi_2])$ and $NL([\Pi_1],[\Pi_2])$ coincide. This turns out to be the case in a neighborhood of $X$ if $X$ is very general on $NL([\Pi_1],[\Pi_2])$, $k>1$ and $\lambda\neq 0,1$. However, there exists a hypersurface $X$ for which $NL([\Pi_1],[\Pi_2])$ is smooth at $X$, but $NL([\Pi_1]+\lambda [\Pi_2])$ is singular for all $\lambda\neq0,1$. We expect that this is due to an embedded component of $NL([\Pi_1]+\lambda[\Pi_2])$. The case $k=1$ was treated before by Dan, in that case $NL([\Pi_1]+\lambda [\Pi_2])$ is nonreduced.

math.AG↗
arXiv2023

On a conjecture on Hodge loci of linear combinations of linear subvarieties

For each $k \geq 5$ we give a counterexample to a conjecture of Movasati on the dimension of certain Hodge loci of cubic hypersurfaces in $\mathbf{P}^{2k+1}$ containing two $k$-planes intersecting in dimension $k-3$. We give similar examples for Hodge loci of cubic hypersurfaces in $\mathbf{P}^{2k+1}$ containing two $k$-planes intersecting in dimension $k-2$ and for quartic hypersurfaces in $\mathbf{P}^{2k+1}$ containing two $k$-planes intersecting in dimension $k-2$. Moreover, we present new evidence for Movasati's conjecture for the values of $k$ for which our type of counterexamples cannot exist, i.e., for $k=3,4$.

math.AG↗
arXiv2022

Deformations of hypersurfaces with non-constant Alexander polynomial

Let X be an irreducible hypersurface in $\mathbb{P}^n$ of degree $d\geq 3$ with only isolated semi-weighted homogeneous singularities, such that $exp(\frac{2πi}{k})$ is a zero of the Alexander polynomial. Then we show that the equianalytic deformation space of $X$ is not $T$-smooth except for a finite list of triples $(n,d,k)$. This result captures the very classical examples by B. Segre of families of degree $6m$ plane curves with $6m^2$, $7m^2$, $8m^2$ and $9m^2$ cusps, where $m\geq 3$. Moreover, we argue that many of the hypersurfaces with non-trivial Alexander polynomial are limits of constructions of hypersurfaces with not $T$-smooth deformation spaces. In many instances this description can be used to construct Alexander-equivalent Zariski pairs.

math.AG↗
arXiv2022

The average Mordell-Weil rank of elliptic surfaces over number fields

Let $K$ be a finitely generated field over $\mathbb{Q}$. Let $\mathcal{X}\to \mathcal{B}$ be a family of elliptic surfaces over $K$ such that each elliptic fibration has the same configuration of singular fibers. Let $r$ be the minimum of the Mordell-Weil rank in this family. Then we show that the locus inside $|\mathcal{B}|$ where the Mordell-Weil rank is at least $r+1$ is a sparse subset. In this way we prove Cowan's conjecture on the average Mordell-Weil rank of elliptic surfaces over $\mathbb{Q}$ and prove a similar result for elliptic surfaces over arbitrary number fields.

math.NT↗
arXiv2020

Infinitesimal Torelli for elliptic surfaces revisited

In this article we give a new proof for the infinitesimal Torelli theorem for minimal elliptic surfaces without multiple fibers with Euler number at least 24 for nonconstant $j$-invariant. In the case of constant $j$-invariant we find a new proof in the case of Euler number at least 72. We also discuss several new counterexamples.

math.AG↗
arXiv2017

Quintic threefolds with triple points

We study the geometry of quintic threefolds $X\subset \mathbb{P}^4$ with only ordinary triple points as singularities. In particular, we show that if a quintic threefold $X$ has a reducible hyperplane section then $X$ has at most $10$ ordinary triple points, and that this bound is sharp. We construct various examples of quintic threefolds with triple points and discuss their defect.

math.AG↗
arXiv2017

Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces

Let $X_λ$ and $X_λ'$ be monomial deformations of two Delsarte hypersurfaces in weighted projective spaces. In this paper we give a sufficient condition so that their zeta functions have a common factor. This generalises results by Doran, Kelly, Salerno, Sperber, Voight and Whitcher [arXiv:1612.09249], where they showed this for a particular monomial deformation of a Calabi-Yau invertible polynomial. It turns out that our factor can be of higher degree than the factor found in [arXiv:1612.09249].

math.NT↗
arXiv2016

Nodal surfaces with obstructed deformations

In this text we show that the deformation space of a nodal surface $X$ of degree $d$ is smooth and of the expected dimension if $d\leq 7$ or $d\geq 8$ and $X$ has at most $4d-5$ nodes. (The case $d\leq 7$ was previously covered by Alexandru Dimca by using different techniques.) For $d\geq 8$ we give explicit examples of nodal surfaces with $4d-4$ nodes, for which the tangent space to the deformation space has larger dimension than expected. We give a short discussion on the shape of the deformation space of surfaces of the form $f_1f_2+f_3^2f_4$, where $f_1$ is a linear form.

math.AG↗
arXiv2015

Mordell-Weil lattices and toric decompositions of plane curves

We extend results of Cogolludo-Agustin and Libgober relating the Alexander polynomial of a plane curve $C$ with the Mordell--Weil rank of certain isotrivial families of jacobians over $\mathbf{P}^2$ of discriminant $C$. In the second part we introduce a height pairing on the $(2,3,6)$ quasi-toric decompositions of a plane curve. We use this pairing and the results in the first part of the paper to construct a pair of degree 12 curves with 30 cusps and Alexander polynomial $t^2-t+1$, but with distinct height pairing. We use the height pairing to show that these curves from a Zariski pair.

math.AG↗
arXiv2015

Nodal complete intersection threefold with defect

In this paper we show that a nodal complete intersection threefold $X$ in $\mathbb{P}^{3+c}$ with defect, but without induced defect, has at least $\sum_{i\leq j} (d_i-1)(d_j-1)$ nodes, provided either $c=2$ or $d_c>\sum_{i=1}^{c-1} d_i$ holds.

math.AG↗
arXiv2015

Chevalley-Weil formula for hypersurfaces in $\mathbf{P}^n$-bundles over curves and Mordell-Weil ranks in function field towers

Let $X$ be a complex hypersurface in a $\mathbf{P}^n$-bundle over a curve $C$. Let $C'\to C$ be a Galois cover with group $G$. In this paper we describe the $\mathbf{C}[G]$-structure of $H^{p,q}(X\times_C C')$ provided that $X\times_C C'$ is either smooth or $n=3$ and $X\times_C C'$ has at most ADE singularities.% and the $\mathbf{C}[G]$-structure of the cohomology of its resolution of singularities. As an application we obtain a geometric proof for an upper bound by Pal for the Mordell-Weil rank of an elliptic surface obtained by a Galois base change of another elliptic surface. If the Galois group of the base field acts trivially on the Galois group of the cover $C'\to C$ then we show that the bound of Pal is weaker than the bound coming from the Shioda-Tate formula.

math.AG↗
arXiv2013

Maximal families of nodal varieties with defect

In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large.

math.AG↗