arXiv subjects
Publications and source records attributed to Yeongrak Kim.
We show that the border rank of the $4 \times 4$ determinant tensor is at least $12$ over $\mathbb{C}$, using the fixed ideal theorem introduced by Buczyńska-Buczyński and the method by Conner-Harper-Landsberg. Together with the known upper bound, this implies that the border rank is exactly $12$.
We investigate new lower bounds on the tensor rank of the determinant and the permanent tensors via recursive usage of the Koszul flattening method introduced by Landsberg-Ottaviani and Hauenstein-Oeding-Ottaviani-Sommese. Our lower bounds on $\mathbf{R} (\det_n)$ completely separate the determinant and the permanent tensors by their tensor ranks. Furthermore, we determine the exact tensor ranks $\mathbf{R} (\det_4) = 12$ and $\mathbf{R} (\operatorname{perm}_4) = 8$ over arbitrary field of characteristic $\neq 2$.
We investigate a geometric criterion for a smooth curve $C$ of genus $14$ and degree $18$ to be described as the zero locus of sections in an Ulrich bundle of rank $3$ on a del Pezzo threefold $V_5 \subset \mathbb{P}^6$. The main challenge is to read off the Pfaffian quadrics defining $V_5$ from geometric structures of $C$. We find that this problem is related to the existence of a special rank-two vector bundle on $C$ with trivial resonance. It gives a description of the image of the rational map that appeared in a work of Ciliberto-Flamini-Knutsen, for the case of degree $5$ del Pezzo threefolds. From an explicit calculation of the Betti table of such a curve, we also deduce the uniqueness of the del Pezzo threefold containing a given curve.
Using the Buchsbaum-Eisenbud structure theorem for a minimal free resolution of an arithmetically Gorenstein variety, we describe a structure theorem for the highest linear syzygies among quadrics defining a del Pezzo variety. Indeed, such syzygies can be represented as columns of a skew-symmetric matrix whose entries are wedge products of linear forms.
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 $\le n-1$.
In this paper, we generalize the geometric mean of two positive definite matrices to that of third-order tensors using the notion of T-product. Specifically, we define the geometric mean of two T-positive definite tensors and verify several properties that "mean" should satisfy including the idempotence and the commutative property, and so on. Moreover, it is shown that the geometric mean is a unique T-positive definite solution of an algebraic Riccati tensor equation and can be expressed as solutions of algebraic Riccati matrix equations. In addition, we investigate the Riemannian manifold associated with the geometric mean for T-positive definite tensors, considering it as a totally geodesic embedded submanifold of the Riemannian manifold associated with the case of matrices. It is particularly shown that the geometric mean of two T-positive definite tensors is the midpoint of a unique geodesic joining the tensors, and the manifold is a Cartan-Hadamard-Riemannian manifold.
The Leray number of an abstract simplicial complex is the minimal integer $d$ where its induced subcomplexes have trivial homology groups in dimension $d$ or greater. We give an upper bound on the Leray number of a complex in terms of how the facets are attached to each other. We also describe the structure of complexes for the equality of the bound that we found. Through the Stanley-Reisner correspondence, our results give an Eisenbud-Goto type inequality for any square-free monomial ideals. This generalizes Terai's result.
In this paper, we present a new formula for the determinant of a $4 \times 4$ matrix. We approach via the sparse optimization problem and derive the formula through the Least Absolute Shrinkage and Selection Operator (LASSO). Our formula has the potential to advance understanding of the algebraic structure of determinants, such as an upper bound of the tensor rank, various notions to measure complexity, and effective computational tools in exterior algebras. We also address several numerical experiments which compare our formula with built-in functions in a computer-algebra system.
In this paper, we suggest a new method for a given tensor to find CP decompositions using a less number of rank $1$ tensors. The main ingredient is the Least Absolute Shrinkage and Selection Operator (LASSO) by considering the decomposition problem as a sparse optimization problem. As applications, we design experiments to find some CP decompositions of the matrix multiplication and determinant tensors. In particular, we find a new formula for the $4 \times 4$ determinant tensor as a sum of $12$ rank $1$ tensors.
In this paper, we present a new formula of the determinant tensor $det_n$ for $n \times n$ matrices. In \cite{kim2023newdet4}, Kim, Ju, and Kim found a new formula of $4 \times 4$ determinant tensor $det_4$ which is available when the base field is not of characteristic $2$. Considering some symmetries in that formula, we found a new formula so that \begin{equation*} \operatorname{Crank}(det_n) \leq \operatorname{rank}(det_n) \leq \frac{n!}{2^{\lfloor(n-2)/2 \rfloor}} \end{equation*} when the base field is not of characteristic $2$.
We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6.
In this paper, we compute an explicit matrix factorization of a rank 9 Ulrich sheaf on a general cubic hypersurface of dimension at most 7, whose existence was proved by Manivel. Instead of using invariant theory, we use Shamash's construction with a cone over the spinor variety. We also describe an algebro-geometric interpretation of our matrix factorization which connects the spinor tenfold and the Cartan cubic.
We discuss the Borisov-Nuer conjecture in connection with the canonical maps from the moduli spaces $\mathcal M_{En,h}^a$of polarized Enriques surfaces with fixed polarization type $h$ to the moduli space $\mathcal F_g$ of polarized $K3$ surfaces of genus $g$ with $g=h^2+1$, and we exhibit a naturally defined locus $Σ_g\subset\mathcal F_g$. One direct consequence of the Borisov-Nuer conjecture is that $Σ_g$ would be contained in a particular Noether-Lefschetz divisor in $\mathcal F_g$, which we call the Borisov-Nuer divisor and we denote by $\mathcal{BN}_g$. In this short note, we prove that $Σ_g\cap\mathcal{BN}_g$ is non-empty whenever $(g-1)$ is divisible by $4$. To this end, we construct polarized Enriques surfaces $(Y, H_Y)$, with $H_Y^2$ divisible by $4$, which verify the conjecture. In particular, the conjecture holds also for any element $\mathcal M_{En,h}^a$, if $h^2$ is divisible by $4$ and $h$ is the same type of polarization.
Using a part of XJC-correspondence by Pirio and Russo, we classify cubic forms $f$ whose Hessian matrices induce matrix factorizations of themselves. When it defines a reduced hypersurface, it satisfies the "secant-singularity" correspondence, that is, it coincides with the secant locus of its singular locus. In particular, when $f$ is irreducible, its singular locus is either one of four Severi varieties.
In this paper, we investigate the existence of Ulrich bundles on a smooth complete intersection of two $4$-dimensional quadrics in $\mathbb P^5$ by two completely different methods. First, we find good ACM curves and use Serre correspondence in order to construct Ulrich bundles, which is analogous to the construction on a cubic threefold by Casanellas-Hartshorne-Geiss-Schreyer. Next, we use Bondal-Orlov's semiorthogonal decomposition of the derived category of coherent sheaves to analyze Ulrich bundles. Using these methods, we prove that any smooth intersection of two 4-dimensional quadrics in $\mathbb P^5$ carries an Ulrich bundle of rank $r$ for every $r \ge 2$. Moreover, we provide a description of the moduli space of stable Ulrich bundles.
In this paper, we prove the existence of an Enriques surface with a polarization of degree four with an Ulrich bundle of rank one. As a consequence, we prove that general polarized Enriques surfaces of degree four, with the same numerical polarization class, carry Ulrich line bundles.
We construct an Ulrich bundle on the blowup at a point when the original variety is embedded by a sufficiently positive linear system and carries an Ulrich bundle. In particular, we describe the relation between special Ulrich bundles on the blown-up surfaces and the original surfaces.
Ulrich bundles are the simplest sheaves from the viewpoint of cohomology tables. Eisenbud and Schreyer conjectured that every projective variety carries an Ulrich bundle, which means it has the same cone of cohomology table as the projective space of same dimension. In this paper we show the existence of stable rank 2 Ulrich bundle on rational surfaces with an anticanonical pencil, under a mild Brill-Noether assumption by using Lazarsfeld-Mukai bundles. Also we see that each of those surfaces carries its Chow form given by the Pfaffian of skew-symmetric morphism coming from an Ulrich bundle.