Searcharxiv⌕ Search

arXiv subjects

Izzet Coskun

Publications and source records attributed to Izzet Coskun.

At least 19 recordsLinked to original sources
arXiv2026

Stability of normal bundles of Brill--Noether curves in $\mathbb{P}^4$

We prove that a general Brill--Noether curve $C$ of genus $g \geq 2$ and degree $d$ in $\mathbb{P}^4$ has stable normal bundle $N_C$ if and only if $$(g, d) \notin \{(2,6), (3,7), (5,8), (6,9), (7,10)\}.$$ Moreover, $N_C$ is strictly semistable if $(g, d) \in \{(3, 7), (5, 8)\}$, and is unstable if $(g, d) \in \{(2, 6), (6, 9), (7, 10)\}$. Our results are valid in any characteristic. Along the way, we also generalize previous results of Larson--Vogt on interpolation for $N_C(-1)$, from characteristic zero to arbitrary characteristic.

math.AG↗
arXiv2026

Obstructions to embedding singular curves in toric varieties

For every integer $d \geq 2$, we show that there exists an irreducible, reduced curve which embeds in $\mathbb{P}^d$ but in no projective normal toric variety of dimension less than $d$. In particular, there exist reduced irreducible curves that embed in $\mathbb{P}^d$ but do not embed in any weighted projective space of dimension less than $d$.

math.AG↗
arXiv2026

Stability of natural bundles on curves

In this paper, we survey recent developments concerning the stability of naturally defined bundles on curves that play a central role in the deformation theory of the curve.

math.AG↗
arXiv2025

Realization of Cohomology Classes in Grassmannians

The study of irreducible subvarieties has recently seen a surge of interest due to connections with convex geometry. In this paper, we study cohomology classes of Grassmannians that are realizable by irreducible subvarieties. We completely classify the cohomology classes that can be realized by irreducible subvarieties in dimensions 2 and 3 and codimensions 2 and 3. We also classify all the classes that can be realizable by an irreducible subvariety for the Grassmannians $G(2,n)$ for $n \leq 6$ and $G(3,6)$. We classify the cohomology classes in all dimensions that are, up to positive multiple, realizable by an irreducible subvariety for the Grassmannians $G(2,n)$.

math.AG↗
arXiv2025

Limit $F$-signature functions of two-variable binomial hypersurfaces

The $F$-signature is a fundamental numerical invariant of singularities in positive characteristic. Its positivity detects strong $F$-regularity, an important class of singularities related to KLT singularities in characteristic zero. In this paper, we compute the limiting $F$-signature function of binomial and other related hypersurfaces in two variables as the characteristic $p \to \infty$. In particular, we show it is a piecewise polynomial function, and relate it to the normalized volume.

math.AC↗
arXiv2024

$\PGL$ orbits on products of flag varieties

In this paper, we study the existence of a dense orbit for the diagonal $\PGL(n)$ action on self-products of partial flag varieties. We determine when there exists a dense orbit for flag varieties of the form $F(k_1, \dots, k_r; n)^m$ when $k_i = i$ or when $m=3$ and $k_i = \ell + i$ for some $\ell \geq 0$. We also show that certain infinite families of products do not have a dense orbit.

math.AG↗
arXiv2024

Brill-Noether theory on the projective plane for bundles with many sections

The Brill-Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill-Noether theory for higher dimensional varieties is less understood. It is hard to determine when Brill-Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. Let $E$ be a semistable sheaf on the projective plane. In this paper, we give an upper bound for $h^0(E)$ in terms of the rank $r$ and the slope $\mu$ of $E$. We show that the bound is achieved precisely when $E$ is a twist of a Steiner bundle. We classify the sheaves $E$ such that $h^0(E)$ is sufficiently close to the upper bound. We determine the nonemptiness, irreducibility and dimension of the Brill-Noether loci in the moduli spaces of sheaves with $h^0(E)$ in this range. When they are nonempty, these Brill-Noether loci are irreducible though almost always of larger than the expected dimension.

math.AG↗
arXiv2024

Weak Brill-Noether on Abelian Surfaces

We study the cohomology of a general stable sheaf on an abelian surface. We say that a moduli space satisfies weak Brill-Noether if the general sheaf has at most one non-zero cohomology group. Let $(X,H)$ be a polarized abelian surface and let $\mathbf{v}=(r,\xi,a)$ be a Mukai vector on $X$ with $\mathbf{v}^2\ge 0$,$r>0$, and $\xi\cdot H>0$. We show that if $\rho(X)=1$ or $\rho(X)=2$ and $X$ contains an elliptic curve, then all the moduli spaces $M_{X,H}(\mathbf{v})$ satisfy weak Brill-Noether. Conversely, if $\rho(X)>2$ or $\rho(X)=2$ and $X$ does not contain an elliptic curve, we show that there are infinitely many moduli spaces $M_{X,H}(\mathbf{v})$ that fail weak Brill-Noether. As a consequence, we classify Chern classes of Ulrich bundles on abelian surfaces.

math.AG↗
arXiv2024

Normal bundles of rational curves in Grassmannians

In projective space over fields of characteristic different from 2, the normal bundle of a general nondegenerate rational curve is balanced. The corresponding statement for rational curves in other Grassmannians can fail. Nevertheless, we prove that the normal bundle of a general rational curve in a Grassmannian decomposes into a direct sum of line bundles whose degrees are at most 2 apart.

math.AG↗
arXiv2023

PGL orbits in tree varieties

In this paper, we introduce tree varieties as a natural generalization of products of partial flag varieties. We study orbits of the PGL action on tree varieties. We characterize tree varieties with finitely many PGL orbits, generalizing a celebrated theorem of Magyar, Weyman and Zelevinsky. We give criteria that guarantee that a tree variety has a dense PGL orbit and provide many examples of tree varieties that do not have dense PGL orbits. We show that a triple of two-step flag varieties $F(k_1, k_2; n)^3$ has a dense PGL orbit if and only if $k_1 + k_2 \not= n$.

math.AG↗
arXiv2023

Generic Beauville's Conjecture

Let $α: X \to Y$ be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under $α$ is semistable if the genus of $Y$ is at least $1$ and stable if the genus of $Y$ is at least $2$. We prove this conjecture if the map $α$ is general in any component of the Hurwitz space of covers of an arbitrary smooth curve $Y$.

math.AG↗
arXiv2023

Stability of normal bundles of Brill-Noether curves

We prove that the normal bundle of a general Brill-Noether curve of genus $g \geq 1$ and degree $d$ in $\mathbb{P}^r$ is semistable if $g=1$ or $g\geq \left \lceil \frac{5r}{2}\right\rceil r(r-1)$, or $d$ is larger than an explicit function of $g$ and $r$. We further prove that the normal bundle is in fact stable if $g\geq 2$ and either $g$ or $d$ satisfy slightly stronger bounds. In particular, for each $r$ and $g\geq 1$ (respectively, $g\geq2$), there are at most finitely many $(d,g)$ for which the normal bundle of the general Brill-Noether curve is not semistable (respectively, stable).

math.AG↗
arXiv2023

Stability of Tschirnhausen Bundles

Let $α: X \to Y$ be a general degree $r$ primitive map of nonsingular, irreducible, projective curves over an algebraically closed field of characteristic zero or larger than $r$. We prove that the Tschirnhausen bundle of $α$ is semistable if $g(Y) \geq 1$ and stable if $g(Y) \geq 2$.

math.AG↗
arXiv2023

Interpolation and moduli spaces of vector bundles on very general blowups of the projective plane

In this paper, we study certain moduli spaces of vector bundles on the blowup of the projective plane in at least 10 very general points. Moduli spaces of sheaves on general type surfaces may be nonreduced, reducible and even disconnected. In contrast, moduli spaces of sheaves on minimal rational surfaces and certain del Pezzo surfaces are irreducible and smooth along the locus of stable bundles. We find examples of moduli spaces of vector bundles on more general blowups of the projective plane that are disconnected and have components of different dimensions. In fact, assuming the SHGH Conjecture, we can find moduli spaces with arbitrarily many components of arbitrarily large dimension.

math.AG↗
arXiv2022

Stability of Normal Bundles of Space Curves

In this paper, we prove that the normal bundle of a general Brill-Noether space curve of degree $d$ and genus $g \geq 2$ is stable if and only if $(d,g) \not\in \{ (5,2), (6,4) \}$. When $g\leq1$ and the characteristic of the ground field is zero, it is classical that the normal bundle is strictly semistable. We show that this fails in characteristic $2$ for all rational curves of even degree.

math.AG↗
arXiv2022

Stability and cohomology of kernel bundles on projective space

In this paper, we study the cohomology of vector bundles on projective space defined as kernels or cokernels of general maps $V_1 \to V_2$, where the $V_i$ are direct sums of line bundles or certain exceptional bundles. We prove an asymptotic cohomology vanishing theorem. We characterize the stability of general Steiner bundles on projective space. We also give a criterion for a Steiner bundle to be ample.

math.AG↗