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Hyunsuk Moon

Publications and source records attributed to Hyunsuk Moon.

9 recordsLinked to original sources
arXiv2024

On the rank index of projective curves of almost minimal degree

In this article, we investigate the rank index of projective curves $\mathscr{C} \subset \mathbb{P}^r$ of degree $r+1$ when $\mathscr{C} = \pi_p (\tilde{\mathscr{C}})$ for the standard rational normal curve $\tilde{\mathscr{C}} \subset \mathbb{P}^{r+1}$ and a point $p \in \mathbb{P}^{r+1} \setminus \tilde{\mathscr{C}}^3$. Here, the rank index of a closed subscheme $X \subset \mathbb{P}^r$ is defined to be the least integer $k$ such that its homogeneous ideal can be generated by quadratic polynomials of rank $\leq k$. Our results show that the rank index of $\mathscr{C}$ is at most $4$, and it is exactly equal to $3$ when the projection center $p$ is a coordinate point of $\mathbb{P}^{r+1}$. We also investigate the case where $p \in \tilde{\mathscr{C}}^3 \setminus \tilde{\mathscr{C}}^2$.

math.AG↗
arXiv2024

Some remarks on the $\mathcal{K}_{p,1}$ Theorem

Let $X$ be a non-degenerate projective irreducible variety of dimension $n \ge 1$, degree $d$, and codimension $e \ge 2$ over an algebraically closed field $\mathbb{K}$ of characteristic $0$. Let $β_{p,q} (X)$ be the $(p,q)$-th graded Betti number of $X$. M. Green proved the celebrating $\mathcal K_{p,1}$-theorem about the vanishing of $β_{p,1} (X)$ for high values for $p$ and potential examples of nonvanishing graded Betti numbers. Later, Nagel-Pitteloud and Brodmann-Schenzel classified varieties with nonvanishing $β_{e-1,1}(X)$. It is clear that $β_{e-1,1}(X) \neq 0$ when there is an $(n+1)$-dimensional variety of minimal degree containing $X$, however, this is not always the case as seen in the example of the triple Veronese surface in $\mathbb{P}^9$. In this paper, we completely classify varieties $X$ with nonvanishing $β_{e-1,1}(X) \neq 0$ such that $X$ does not lie on an $(n+1)$-dimensional variety of minimal degree. They are exactly cones over smooth del Pezzo varieties whose Picard number is $\le n-1$.

math.AG↗
arXiv2023

On the rank index of some quadratic varieties

Regarding the generating structure of the homogeneous ideal of a projective variety $X \subset \mathbb{P}^r$, we define the rank index of $X$ to be the smallest integer $k$ such that $I(X)$ can be generated by quadratic polynomials of rank at most $k$. Recently it is shown that every Veronese embedding has rank index $3$ if the base field has characteristic $\ne 2, 3$. In this paper, we introduce some basic ways of how to calculate the rank index and find its values when $X$ is some other classical projective varieties such as rational normal scrolls, del Pezzo varieties, Segre varieties and the Plücker embedding of the Grassmannian of lines.

math.AG↗
arXiv2023

A family of explicit Waring decompositions of a polynomial

In this paper we settle some polynomial identity which provides a family of explicit Waring decompositions of any monomial $X_0^{a_0}X_1^{a_1}\cdots X_n^{a_n}$ over a field $\Bbbk$. This gives an upper bound for the Waring rank of a given monomial and naturally leads to an explicit Waring decomposition of any homogeneous form and, eventually, of any polynomial via (de)homogenization. Note that such decomposition is very useful in many applications dealing with polynomial computations, symmetric tensor problems and so on. We discuss some computational aspect of our result as comparing with other known methods and also present a computer implementation for potential use in the end.

math.AC↗
arXiv2022

A new bound for the Real Waring rank of monomials

In this paper we consider the Waring rank of monomials over the real and the rational numbers. We give a new upper bound for it by establishing a way in which one can take a structured apolar set for any given monomial $X_0^{a_0}X_1^{a_1}\cdots X_n^{a_n}$ ($a_i>0$). This bound coincides with the real Waring rank in the case $n=1$ and in the case $\min(a_i)=1$, which are all the known cases for the real rank of monomials. Our bound is also lower than any other known general bounds for the real Waring rank. Since all of the constructions are still valid over the rational numbers, this provides a new result for the rational Waring rank of any monomial as well. Some examples and computational implementation for potential use are presented in the end.

math.AG↗
arXiv2021

Rank 3 Quadratic Generators of Veronese Embeddings

Let $L$ be a very ample line bundle on a projective scheme $X$ defined over an algebraically closed field $\Bbbk$ with ${\rm char}~\Bbbk \neq 2$. We say that $(X,L)$ satisfies property $\mathsf{QR}(k)$ if the homogeneous ideal of the linearly normal embedding $X \subset \mathbb{P}H^0 (X,L)$ can be generated by quadrics of rank $\leq k$. Many classical varieties such as Segre-Veronese embeddings, rational normal scrolls and curves of high degree satisfy property $\mathsf{QR}(4)$. In this paper, we first prove that if ${\rm char}~\Bbbk \neq 3$ then $(\mathbb{P}^n , \mathcal{O}_{\mathbb{P}^n} (d))$ satisfies property $\mathsf{QR}(3)$ for all $n \geq 1$ and $d \geq 2$. We also investigate an asymptotic behavior of property $\mathsf{QR}(3)$ for any projective scheme. Namely, we prove that $(i)$ if $X \subset \mathbb{P} H^0 (X,L)$ is $m$-regular then $(X,L^d )$ satisfies property $\mathsf{QR}(3)$ for all $d \geq m$ and $(ii)$ if $A$ is an ample line bundle on $X$ then $(X,A^d )$ satisfies property $\mathsf{QR}(3)$ for all sufficiently large even number $d$. These results provide an affirmative evidence for the expectation that property $\mathsf{QR}(3)$ holds for all sufficiently ample line bundles on $X$, as in the cases of Green-Lazarsfeld's condition $\mathrm{N}_p$ and Eisenbud-Koh-Stillman's determininantal presentation in [EKS88]. Finally, when ${\rm char}~\Bbbk = 3$ we prove that $(\mathbb{P}^n , \mathcal{O}_{\mathbb{P}^n} (2))$ fails to satisfy property $\mathsf{QR}(3)$ for all $n \geq 3$.

math.AG↗
arXiv2016

Spaces of Sums of Powers and Real Rank Boundaries

We investigate properties of Waring decompositions of real homogeneous forms. We study the moduli of real decompositions, so-called Space of Sums of Powers, naturally included in the Variety of Sums of Powers. Explicit results are obtained for quaternary quadrics, relating the algebraic boundary of ${\rm SSP}$ to various loci in the Hilbert scheme of four points in $\mathbb{P}^3$. Further, we study the locus of general real forms whose real rank coincides with the complex rank. In case of quaternary quadrics the boundary of this locus is a degree forty hypersurface $J(σ_3(v_3(\mathbb{P}^3)),τ(v_3(\mathbb{P}^3)))$.

math.AG↗
arXiv2016

Real Rank Geometry of Ternary Forms

We study real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained for quadrics and cubics. For quintics we determine the real rank boundary: it is a hypersurface of degree 168. For quartics, sextics and septics we identify some of the components of the real rank boundary. The real varieties of sums of powers are stratified by discriminants that are derived from hyperdeterminants.

math.AG↗