N. Ilten

activity

20082026

most citedToric Degenerations of Fano Threefolds Giving Weak Landau-Ginzburg Models

44 citations · 315 across the 37 of their papers we have counts for

collaborators

43 papers

math.AG2026

Higher cotangent cohomology for Stanley-Reisner rings

Nathan Ilten, Francesco Meazzini, Andrea Petracci

Inspired by work of Altmann and Christophersen, we study the graded pieces of the cotangent cohomology $T_{S_{K}}^{i}$ , $i \geq 3$ of the Stanley-Reisner ring $S_{\mathcal{K…

math.AG2025

Local Equations for Hilbert Schemes of Points

Nathan Ilten, Francesco Meazzini, Andrea Petracci

We compute the completion of the local ring of the Hilbert scheme of degree $n + 1$ subschemes of $A^{n}$ at the point corresponding to the ideal $\langle x_1,\ldots,x_n\rangl…

math.AC2025

Gröbner Cones for Finite Type Cluster Algebras

Nathan Ilten, Karolyn So

Let $A$ be a cluster algebra of finite cluster type. We study the Gröbner cone $C_{A}$ parametrizing term orders inducing an initial degeneration of t…

math.AG2024

Locally Trivial Deformations of Toric Varieties

Nathan Ilten, Sharon Robins

We study locally trivial deformations of toric varieties from a combinatorial point of view. For any fan $Σ$ , we construct a deformation functor $Def_{Σ}$ by considering Čec…

math.AG2023

Rational Curves in Projective Toric Varieties

Nathan Ilten, Jake Levinson

We study embedded rational curves in projective toric varieties. Generalizing results of the first author and Zotine for the case of lines, we show that any degree $d$ rational cur…

math.AG2023

Local Euler characteristics of $A_{n}$ -singularities and their application to hyperbolicity

Nils Bruin, Nathan Ilten, Zhe Xu

Wahl's local Euler characteristic measures the local contributions of a singularity to the usual Euler characteristic of a sheaf. Using tools from toric geometry, we study the loca…

math.AG2022

Toric Degenerations of Low Degree Hypersurfaces

Nathan Ilten, Oscar Lautsch

We show that a sufficiently general hypersurface of degree $d$ in $n$ -dimensional projective space admits a toric Gröbner degeneration if and only if $d \leq 2 n - 1$ .

math.AG2021★ 1 cited

Deformation Theory for Finite Cluster Complexes

Nathan Ilten, Alfredo Nájera Chávez, Hipolito Treffinger

We study the deformation theory of the Stanley-Reisner rings associated to cluster complexes for skew-symmetrizable cluster algebras of geometric and finite cluster type. In partic…

math.AG2021

Tropical tangents for complete intersection curves

Nathan Ilten, Yoav Len

We consider the tropicalization of tangent lines to a complete intersection curve $X$ in $P^{n}$ . Under mild hypotheses, we describe a procedure for computing the tropicaliz…

math.AG2021

Khovanskii-finite rational curves of arithmetic genus 2

Nathan Ilten, Ahmad Mokhtar

We study the existence of Khovanskii-finite valuations for rational curves of arithmetic genus two. We provide a semi-explicit description of the locus of degree $n + 2$ rational cur…

math.AG2019

Fano Schemes of Complete Intersections in Toric Varieties

Nathan Ilten, Tyler L. Kelly

We study Fano schemes $F_{k} (X)$ for complete intersections $X$ in a projective toric variety $Y \subset P^{n}$ . Our strategy is to decompose $F_{k} (X)$ into closed subschemes ba…

math.AG2019

Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

Jarosław Buczyński, Nathan Ilten, Emanuele Ventura

In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occu…

math.AG2019

Algebraic Hyperbolicity for Surfaces in Toric Threefolds

Christian Haase, Nathan Ilten

Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We il…

math.AG2018★ 5 cited

Deformations of Smooth Complete Toric Varieties: Obstructions and the Cup Product

Nathan Ilten, Charles Turo

Let $X$ be a complete $Q$ -factorial toric variety. We explicitly describe the space $H^{2} (X, T_{X})$ and the cup product map $H^{1} (X, T_{X}) \times H^{1} (X, T_{X}) \to H^{2} (X, T_{X})$ in com…

math.AG2018★ 3 cited

Projective duals to algebraic and tropical hypersurfaces

Nathan Ilten, Yoav Len

We study a tropical analogue of the projective dual variety of a hypersurface. When $X$ is a curve in $P^{2}$ or a surface in $P^{3}$ , we provide an explicit descript…

math.AG2017

Fujita's Freeness Conjecture for $T$ -Varieties of Complexity One

Klaus Altmann, Nathan Ilten

We prove Fujita's freeness conjecture for Gorenstein complexity-one $T$ -varieties with rational singularities.

math.AG2017★ 1 cited

Classifying Fano Complexity-One $T$ -Varieties via Divisorial Polytopes

Nathan Ilten, Marni Mishna, Charlotte Trainor

The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and Süß to a correspondence between Gorenstein Fano complexity-one…

math.AG2017★ 11 cited

Rational Complexity-One T-Varieties are Well-Poised

Nathan Ilten, Christopher Manon

Given an affine rational complexity-one $T$ -variety $X$ , we construct an explicit embedding of $X$ in affine space $A^{n}$ . We show that this embedding is well-poised, that…

math.AG2016

Fano Schemes for Generic Sums of Products of Linear Forms

Nathan Ilten, Hendrik Süß

We study the Fano scheme of $k$ -planes contained in the hypersurface cut out by a generic sum of products of linear forms. In particular, we show that under certain hypotheses, lin…

math.AG2016★ 4 cited

On Fano Schemes of Toric Varieties

Nathan Ilten, Alexandre Zotine

Let $X_{A}$ be the projective toric variety corresponding to a finite set of lattice points $A$ . We show that irreducible components of the Fano scheme $\mathbf{F…

math.AG2015★ 3 cited

Unobstructed Stanley-Reisner Degenerations for Dual Quotient Bundles on $G (2, n)$

Nathan Ilten, Charles Turo

Let $Q^{*}$ denote the dual of the quotient bundle on the Grassmannian $G (2, n)$ . We prove that the ideal of $Q^{*}$ in its natural embedding has initial ideal equal to the Stanley-Reis…

math.AG2015★ 32 cited

K-Stability for Fano Manifolds with Torus Action of Complexity One

Nathan Ilten, Hendrik Süß

We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension one. Using a recent result of Datar and Szekelyhidi, we effectively determine the e…

math.AG2015★ 12 cited

Product Ranks of the $3 \times 3$ Determinant and Permanent

Nathan Ilten, Zach Teitler

We show that the product rank of the $3 \times 3$ determinant is $5$ , and the product rank of the $3 \times 3$ permanent is $4$ . As a corollary, we obtain that the tensor ranks of th…

math.AG2015★ 2 cited

F-Split and F-Regular Varieties with a Diagonalizable Group Action

Piotr Achinger, Nathan Ilten, Hendrik Süß

Let $H$ be a diagonalizable group over an algebraically closed field $k$ of positive characteristic, and $X$ a normal $k$ -variety with an $H$ -action. Under a mild hypothesis, e.g.…

Higher cotangent cohomology for Stanley-Reisner rings

Local Equations for Hilbert Schemes of Points

Gröbner Cones for Finite Type Cluster Algebras

Locally Trivial Deformations of Toric Varieties

Rational Curves in Projective Toric Varieties

Local Euler characteristics of An​-singularities and their application to hyperbolicity

Toric Degenerations of Low Degree Hypersurfaces

Deformation Theory for Finite Cluster Complexes

Tropical tangents for complete intersection curves

Khovanskii-finite rational curves of arithmetic genus 2

Fano Schemes of Complete Intersections in Toric Varieties

Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

Algebraic Hyperbolicity for Surfaces in Toric Threefolds

Deformations of Smooth Complete Toric Varieties: Obstructions and the Cup Product

Projective duals to algebraic and tropical hypersurfaces

Fujita's Freeness Conjecture for T-Varieties of Complexity One

Classifying Fano Complexity-One T-Varieties via Divisorial Polytopes

Rational Complexity-One T-Varieties are Well-Poised

Fano Schemes for Generic Sums of Products of Linear Forms

On Fano Schemes of Toric Varieties

Unobstructed Stanley-Reisner Degenerations for Dual Quotient Bundles on G(2,n)

K-Stability for Fano Manifolds with Torus Action of Complexity One

Product Ranks of the 3×3 Determinant and Permanent

F-Split and F-Regular Varieties with a Diagonalizable Group Action

Local Euler characteristics of $A_{n}$ -singularities and their application to hyperbolicity

Fujita's Freeness Conjecture for $T$ -Varieties of Complexity One

Classifying Fano Complexity-One $T$ -Varieties via Divisorial Polytopes

Unobstructed Stanley-Reisner Degenerations for Dual Quotient Bundles on $G (2, n)$

Product Ranks of the $3 \times 3$ Determinant and Permanent