arXiv subjects

Sümeyra Sakallı

Publications and source records attributed to Sümeyra Sakallı.

12 recordsLinked to original sources

arXiv2026

Xiao's Degree-Four Fibration and the Polizzi Model

Let $S_E$ be the surface in Xiao's degree-four family identified by Polizzi as a smooth divisor in $E(3)$. Starting from Polizzi's branch model and Xiao's classification of the thirteen singular fibers, we derive the local spherical braids and their Picard--Lefschetz lifts, including the colored $3+3$ partitions at the seven reducible fibers. We identify the normalization of each bisection with $E/\{\pm1\}$ and its degree-two ruling map with the quotient by the involution induced by translation by a nonzero two-torsion point. We prove $π_1(S_E)\cong\mathbb Z^2$, compute the elementary-divisor-$4$ Albanese kernel of a regular genus-two fiber, and show that the six nonseparating Picard--Lefschetz transformations represent the six cusps of $Γ(4)$. We also prove that the natural product tori near an elliptic section are nullhomologous and cannot lower $b_1$ below $2$ by torus surgery. As a separate application of the degree-three monodromy, we give a twisted-double construction of an exotic $\mathbb CP^2\#7\overline{\mathbb CP}^{\,2}$.

math.AG↗

arXiv2026

Gurtas Lefschetz Fibrations, Rational Blowdowns, and Exotic Symplectic Four-Manifolds

We study negative spheres obtained from exceptional sections of Gurtas Lefschetz fibrations. Starting with the known system of $4n$ disjoint $(-1)$-sections, we show that a marked double sum contains $4n$ symplectic $(-2)$-spheres and $2n$ smooth $(-4)$-spheres obtained by pairwise tubing. By carrying the same sections through a fourfold sum, we instead obtain $4n$ disjoint symplectic $(-4)$-spheres and hence symplectic rational blowdowns. We then consider the Lefschetz fibrations on knot-surgered elliptic surfaces. Their branched-cover description gives $2n$ branch sections together with simultaneous geometric duals arising from node resolution. In the knot-surgery double these sections glue to symplectic $(-4)$-spheres, while the duals make the complement of every subcollection simply connected. The resulting rational blowdowns provide simply connected exotic symplectic four-manifolds under the parity condition stated in the main theorem. We also record the boundary-multitwist relation determined by the Gurtas sections.

math.GT↗

arXiv2026

Construction and Periodicity of Minimal Resolution Graphs of Suspension Singularities

We develop computational tools for constructing the minimal resolution graphs of suspension singularities. In particular, we provide a Magma program that implements Nemethi's algorithm that can be used to construct the minimal resolution graph of any suspension singularity. The program outputs the resulting resolution graph as structured graph data, encoding the vertices, edges, and decorations of the graph. We further develop a SageMath program that takes this data as input and produces graphical representations of the resolution graphs, facilitating their visualization and analysis. For three families of suspension singularities, we prove that the corresponding minimal resolution graphs are periodic, with periods equal to the orders of the corresponding monodromy factorizations. We also provide Magma programs for generating compactified resolution graphs of suspension singularities and checking graph isomorphisms over a specified range of parameters.

math.GT↗

arXiv2024

The contact cut graph and a Weinstein $\mathcal{L}$-invariant

We define and study the contact cut graph which is an analogue of Hatcher and Thurston's cut graph for contact geometry, inspired by contact Heegaard splittings. We show how oriented paths in the contact cut graph correspond to Lefschetz fibrations and multisection with divides diagrams. We also give a correspondence for achiral Lefschetz fibrations. We use these correspondences to define a new invariant of Weinstein domains, the Weinstein $\mathcal{L}$-invariant, that is a symplectic analogue of the Kirby-Thompson's $\mathcal{L}$-invariant of smooth $4$-manifolds. We discuss the relation of Lefschetz stabilization with the Weinstein $\mathcal{L}$-invariant. We present topological and geometric constraints of Weinstein domains with $\mathcal{L}=0$. We also give two families of examples of multisections with divides that have arbitrarily large $\mathcal{L}$-invariant.

math.GT↗

arXiv2023

Splitting Algebraic Singular Fibrations via Perturbation of Branch Covers

In a previous paper \cite{SV}, the authors studied the isolated singular fibers that can occur in algebraic fibrations of certain genus two fibrations. There the goal was to determine their monodromy factorizations with the goal of determining a dictionary between a set of curve configurations and certain words in the mapping class group. Each such curve configuration was originally cataloged by Namikawa and Ueno \cite{NamikawaUeno-list} in their list of genus two fibrations. We studied four families of polynomials, we restricted to fibrations whose fibers have boundary, considering the isolated affine singularity referenced in \cite{NamikawaUeno-list}. We resolved the singularities and, using carefully chosen perturbations, deformed them into Lefschetz fibrations and determined their monodromy factorizations. In two of those families, we also obtained strong information about how the central fiber compactifies in a fibration with closed fibers. In this paper we work on the other two cases by recreating the singular fibers using a different family of polynomials. In \cite{SV}, all the algebraic curves in the fibrations were given expressly as hyperelliptic equations. We utilized this symmetry both to construct the deformation and to recover the monodromy factorization. In this paper the curves are no longer expressly hyperelliptic--the quotient curve is no longer just $\CC$, the branch curves are no longer embedded in $\CC^2$, and the fibrations utilized in the quotient are now more complicated. We do, though, recover the desired behavior of the compactification of the central fiber, its deformation to a Lefschetz fibration, and the corresponding monodromy factorization.

math.GT↗

arXiv2023

Singular fibers in algebraic fibrations of genus 2 and their monodromy factorizations

Kodaira's classification of singular fibers in elliptic fibrations and its translation into the language of monodromies and Lefschetz fibrations has been a boon to the study of 4-manifolds. In this article, we begin the work of translating between singular fibers of genus 2 families of algebraic curves and the positive Dehn twist factorizations of Lefschetz fibrations for a certain subset of the singularities described by Namikawa and Ueno in the 70s. We look at four families of hypersurface singularities in $\mathbb{C}^3$. Each hypersurface comes equipped with a fibration by genus 2 algebraic curves which degenerate into a single singular fiber. We determine the resolution of each of the singularities in the family and find a flat deformation of the resolution into simpler pieces, resulting in a fibration of Lefschetz type. We then record the description of the Lefschetz as a positive factorization in Dehn twists. This gives us a dictionary between configurations of curves and monodromy factorizations for some singularities of genus 2 fibrations.

math.GT↗

arXiv2021

Slicing knots in definite 4-manifolds

We study the $\mathbb{CP}^2$-slicing number of knots, i.e. the smallest $m\geq 0$ such that a knot $K\subseteq S^3$ bounds a properly embedded, null-homologous disk in a punctured connected sum $(\#^m\mathbb{CP}^2)^{\times}$. We give a lower bound on the smooth $\mathbb{CP}^2$-slicing number of a knot in terms of its double branched cover, and we find knots with arbitrarily large but finite smooth $\mathbb{CP}^2$-slicing number. We also give an upper bound on the topological $\mathbb{CP}^2$-slicing number in terms of the Seifert form and find knots for which the smooth and topological $\mathbb{CP}^2$-slicing numbers are both finite, nonzero, and distinct.

math.GT↗

arXiv2021

Symplectic 4-Manifolds on the Noether Line and between the Noether and Half Noether Lines

We construct simply connected, minimal, symplectic 4-manifolds with exotic smooth structures and each with one Seiberg-Witten basic class up to sign, on the Noether line and between the Noether and half Noether lines by star surgeries introduced by Karakurt and Starkston, and by using complex singularities. We also construct certain configurations of complex singularities in the rational elliptic surfaces geometrically, without using any monodromy arguments. By using these configurations, we give symplectic embeddings of star shaped plumbings inside (some blow-ups of) elliptic surfaces.

math.GT↗

arXiv2021

Complex Ball Quotients and New Symplectic 4-manifolds with Nonnegative Signatures

We present the various constructions of new symplectic $4$-manifolds with non-negative signatures using the complex surfaces on the BMY line $c_1^2 = 9χ_h$, the Cartwright-Steger surfaces, the quotients of Hirzebruch's certain line-arrangement surfaces, along with the exotic symplectic $4$-manifolds constructed in \cite{AP2, AS}. In particular, our constructions yield to (i) an irreducible symplectic and infinitely many non-symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to $(2n-1)CP^{2}\#(2n-1)\bar{CP}^{2}$ for each integer $n \geq 9$, (ii) the families of simply connected irreducible nonspin symplectic $4$-manifolds that have the smallest Euler characteristics among the all known simply connected $4$-manifolds with positive signatures and with more than one smooth structure. We also construct a complex surface with positive signature from the Hirzebruch's line-arrangement surfaces, which is a ball quotient.

math.SG↗

arXiv2021

Generalized Chain Surgeries and Applications

We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration.

math.GT↗

arXiv2019

On the BV structure on the cohomology of moduli space

The question of vanishing of the BV operator on the cohomology of the moduli space of Riemann surfaces is investigated. The BV structure, which comprises a BV operator and an antibracket, is identified, vanishing theorems are proven, and a counterexample is provided.

math.AG↗

arXiv2015

On the geography of simply connected nonspin symplectic $4$-manifolds with nonnegative signature

In \cite{AP3, AHP}, the first author and his collaborators constructed the irreducible symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to $(2n-1){\mathbb{CP}}^{2}\#(2n-1)\overline{\mathbb{CP}}^{2}$ for each integer $n \geq 25$, and the families of simply connected irreducible nonspin symplectic $4$-manifolds with positive signature that are interesting with respect to the symplectic geography problem. In this paper, we improve the main results in \cite{AP3, AHP}. In particular, we construct (i) an infinitely many irreducible symplectic and non-symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to $(2n-1){\mathbb{CP}^{2}}\#(2n-1)\overline{\mathbb{CP}}^{2}$ for each integer $n \geq 12$, and (ii) the families of simply connected irreducible nonspin symplectic $4$-manifolds that have the smallest Euler characteristics among the all known simply connected $4$-manifolds with positive signature and with more than one smooth structure. Our construction uses the complex surfaces of Hirzebruch and Bauer-Catanese on Bogomolov-Miyaoka-Yau line with $c_1^2 = 9χ_h = 45$, along with the exotic symplectic $4$-manifolds constructed in \cite{A4, AP1, ABBKP, AP2, AS}.

math.GT↗