arXiv subjects

Junho Choe

Publications and source records attributed to Junho Choe.

6 recordsLinked to original sources

arXiv2025

Quadratic equations of tangent varieties via four-way tensors of linear forms

In the present paper we construct quadratic equations and linear syzygies for tangent varieties using 4-way tensors of linear forms and generalize this method to higher secant varieties of higher osculating varieties. Such equations extend the classical determinantal ones of higher secant varieties and span all the equations of the same degree for smooth projective curves completely embedded by sufficiently positive line bundles, proving a variant of the Eisenbud-Koh-Stillman conjecture on determinantal equations. On the other hand, our syzygies are compatible with the Green-Lazarsfeld classes and generate the corresponding Koszul cohomology groups for Segre varieties with a prescribed number of factors. To obtain these results we describe the equations of minimal possible degrees and reinterpret the Green-Lazarsfeld classes from the perspective of representation theory.

math.AG↗

arXiv2025

A Fr\"{o}berg type theorem for higher secant complexes

We generalize the celebrated Fr\"{o}berg's theorem to embedded joins of copies of a simplicial complex, namely higher secant complexes to the simplicial complex, in terms of property $N_{q+1,p}$ due to Green and Lazarsfeld. Furthermore, we investigate combinatorial phenomena parallel to geometric ones observed for higher secant varieties of minimal degree.

math.AC↗

arXiv2023

Syzygies of secant varieties of smooth projective curves and gonality sequences

The purpose of this paper is to prove that one can read off the gonality sequence of a smooth projective curve from syzygies of secant varieties of the curve embedded by a line bundle of sufficiently large degree. More precisely, together with Ein-Niu-Park's theorem, our main result shows that the gonality sequence of a smooth projective curve completely determines the shape of the minimal free resolutions of secant varieties of the curve of sufficiently large degree. This is a natural generalization of the gonality conjecture on syzygies of smooth projective curves established by Ein-Lazarsfeld and Rathmann to the secant varieties.

math.AG↗

arXiv2022

Determinantal characterization of higher secant varieties of minimal degree

A variety of minimal degree is one of the basic objects in projective algebraic geometry and has been classified and characterized in many aspects. On the other hand, there are also minimal objects in the category of higher secant varieties, and their algebraic and geometric structures seem to share many similarities with those of varieties of minimal degree. We prove in this paper that higher secant varieties of minimal degree have determinantal presentation of two types, i.e. scroll type and Veronese type. Our result generalizes the del Pezzo-Bertini classification for varieties of minimal degree. Also, as a consequence, we show that for any smooth projective variety having higher secant variety of minimal degree, the embedding line bundle admits a special decomposition into two line bundles as so do those of the well-known examples: varieties of minimal degree, smooth del Pezzo varieties, Segre varieties and 2-Veronese varieties.

math.AG↗

arXiv2022

Castelnuovo-Mumford regularity of unprojections and the Eisenbud-Goto regularity conjecture

McCullough and Peeva found sequences of counterexamples to the Eisenbud--Goto conjecture on the Castelnuovo--Mumford regularity by using Rees-like algebras, where entries of each sequence have increasing dimensions and codimensions. In this paper we suggest another method to construct counterexamples to the conjecture with any fixed dimension $n\geq3$ and any fixed codimension $e\geq2$. Our strategy is an unprojection process and utilizes the possible complexity of homogeneous ideals with three generators. Furthermore, our counterexamples exhibit how singularities affect the Castelnuovo--Mumford regularity.

math.AG↗

arXiv2021

A matryoshka structure of higher secant varieties and the generalized Bronowski's conjecture

In projective algebraic geometry, there are classical and fundamental results that describe the structure of geometry and syzygies, and many of them characterize varieties of minimal degree and del Pezzo varieties. In this paper, we consider analogous objects in the category of higher secant varieties. Our main theorems say that there is a matryoshka structure among those basic objects including a generalized $K_{p,1}$ theorem, syzygetic and geometric characterizations of higher secant varieties of minimal degree and del Pezzo higher secant varieties, defined in this paper. For our purpose, we prove a weak form of the generalized Bronowski's conjecture raised by C. Ciliberto and F. Russo that relates the identifiability for higher secant varieties to the geometry of tangential projections.

math.AG↗