arXiv subjects
Publications and source records attributed to Kangjin Han.
White's conjecture predicts quadratic generators for the ideal of any matroid base polytope. We prove that White's conjecture for any matroid $M$ implies it also for any matroid $M'$, where $M$ and $M'$ differ by one basis. Our study is motivated by inner projections of algebraic varieties.
In this paper, we study minimal generators of the (saturated) defining ideal of $\sigma_k(v_d(\mathbb{P}^n))$ in $\mathbb{P}^{N}$ with ${N=\binom{n+d}{d}-1}$, the $k$-secant variety of $d$-uple Veronese embedding of projective $n$-space, of a relatively small degree. We first show that the prime ideal $I(\sigma_4(v_3(\mathbb{P}^3)))$ can be minimally generated by 36 homogeneous polynomials of degree $5$. It implies that $\sigma_4(v_3(\mathbb{P}^3)) \subset \mathbb{P}^{19}$ is a del Pezzo $4$-secant variety (i.e., $\mathrm{deg}(\sigma_4(v_3(\mathbb{P}^3))) = 105$ and the sectional genus $\pi(\sigma_4(v_3(\mathbb{P}^3))) = 316$) and provides a new example of an arithmetically Gorenstein variety of codimension $4$. As an application, we decide non-singularity of a certain locus in $\sigma_4(v_3(\mathbb{P}^3))$. By inheritance, generators of $I(\sigma_4(v_3(\mathbb{P}^n)))$ are also obtained for any $n \geq 3$. We also propose a procedure to compute the first non-trivial degree piece $I(\sigma_k(v_d(\mathbb{P}^n)))_{k+1}$ for a general $k$-th secant case, in terms of prolongation and weight space decomposition, based on the method used for $\sigma_4(v_3(\mathbb{P}^3))$ and treat a few more cases of $k$-secant varieties of the Veronese embedding of a relatively small degree in the end.
In this paper, we settle a conjecture due to Sturmfels and Uhler concerning generation of the prime ideal of the variety associated to the Gaussian graphical model of any cycle graph. Our methods are general and applicable to a large class of ideals with radical initial ideals.
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.
The $k$-th secant variety of a projective variety $X \subset \mathbb{P}^N$, denoted by $\sigma_k(X)$, is defined to be the closure of the union of $(k-1)$-planes spanned by $k$ points on $X$. In this paper, we examine the $k$-th secant variety $\sigma_k(v_d(\mathbb{P}^n)) \subset \mathbb{P}^N$ of the image of the $d$-uple Veronese embedding $v_d$ of $\mathbb{P}^n$ to $\mathbb{P}^N$ with $N=\binom{n+d}{d}-1$, and focus on the singular locus of $\sigma_k(v_d(\mathbb{P}^n))$, which is only known for $k\le3$. To study the singularity for arbitrary $k,d,n$, we define \emph{the $m$-subsecant locus} of $\sigma_k(v_d(\mathbb{P}^n))$ to be the union of $\sigma_k(v_d(\mathbb{P}^m))$ with any $m$-plane $\mathbb{P}^m \subset \mathbb{P}^n$. By investigating the projective geometry of moving embedded tangent spaces along subvarieties and using known results on the secant defectivity and the identifiability of symmetric tensors, we determine whether the $m$-subsecant locus is contained in the singular locus of $\sigma_k(v_d(\mathbb{P}^n))$ or not. Depending on the value of $k$, these subsecant loci show an interesting trichotomy between generic smoothness, non-trivial singularity, and trivial singularity. In many cases, they can be used as a new source for the singularity of the $k$-th secant variety of $v_d(\mathbb{P}^n)$ other than the trivial one, the $(k-1)$-th secant variety of $v_d(\mathbb{P}^n)$. We also consider the case of the $4$-th secant variety of $v_d(\mathbb{P}^n)$ by applying main results and computing conormal space via a certain type of Young flattening. Finally, we present some generalizations and discussions for further developments.
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.
Let $X\subset\mathbb{P}^{n+e}$ be any $n$-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: one is the property $\mathbf{N}_{d,p}~(d\ge 2, ~p\geq 1)$, which means that $X$ is $d$-regular up to $p$-th step in the minimal free resolution and the other is a new notion $\mathrm{ND}(\ell)$ which generalizes the classical "being nondegenerate" to the condition that requires a general finite linear section not to be contained in any hypersurface of degree $\ell$. First, we introduce condition $\mathrm{ND}(\ell)$ and consider examples and basic properties deduced from the notion. Next we prove sharp upper bounds on the graded Betti numbers of the first non-trivial strand of syzygies, which generalize results in the quadratic case to higher degree case, and provide characterizations for the extremal cases. Further, after regarding some consequences of property $\mathbf{N}_{d,p}$, we characterize the resolution of $X$ to be $d$-linear arithmetically Cohen-Macaulay as having property $\mathbf{N}_{d,e}$ and condition $\mathrm{ND}(d-1)$ at the same time. From this result, we obtain a syzygetic rigidity theorem which suggests a natural generalization of syzygetic rigidity on $2$-regularity due to Eisenbud-Green-Hulek-Popescu to a general $d$-regularity.
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 $n \geq 3$.
In this paper we study singularities of third secant varieties of Veronese embedding $v_d(\mathbb{P}^n)$, which corresponds to the variety of symmetric tensors of border rank at most three in $(\mathbb{C}^{n+1})^{\otimes d}$.
Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X. Let W_k be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci W_k for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1,3,3), and curves. An intermediate result provides a lower bound on the dimension of any GL_n orbit of a homogeneous form.
In this paper we study graded Betti numbers of any nondegenerate 3-regular algebraic set $X$ in a projective space $\mathbb P^{n}$. More concretely, via Generic initial ideals (Gins) method we mainly consider `tailing' Betti numbers, whose homological index is not less than $\mathrm{codim}(X,\mathbb P^{n})$. For this purpose, we first introduce a key definition `$\mathrm{ND(1)}$ property', which provides a suitable ground where one can generalize the concepts such as `being nondegenerate' or `of minimal degree' from the case of varieties to the case of more general closed subschemes and give a clear interpretation on the tailing Betti numbers. Next, we recall basic notions and facts on Gins theory and we analyze the generation structure of the reverse lexicographic (rlex) Gins of 3-regular $\mathrm{ND(1)}$ subschemes. As a result, we present exact formulae for these tailing Betti numbers, which connect them with linear normality of general linear sections of $X\cap Λ$ with a linear subspace $Λ$ of dimension at least $\mathrm{codim}(X,\mathbb P^{n})$. Finally, we consider some applications and related examples.
In the present paper, we consider upper bounds of higher linear syzygies i.e. graded Betti numbers in the first linear strand of the minimal free resolutions of projective varieties in arbitrary characteristic. For this purpose, we first remind `Partial Elimination Ideals (PEIs)' theory and introduce a new framework in which one can study the syzygies of embedded projective schemes well using PEIs theory and the reduction method via inner projections. Next we establish fundamental inequalities which govern the relations between the graded Betti numbers in the first linear strand of an algebraic set $X$ and those of its inner projection $X_q$. Using these results, we obtain some natural sharp upper bounds for higher linear syzygies of any nondegenerate projective variety in terms of the codimension with respect to its own embedding and classify what the extremal case and next-to-extremal case are. This is a generalization of Castelnuovo and Fano's results on the number of quadrics containing a given variety and another characterization of varieties of minimal degree and del Pezzo varieties from the viewpoint of `syzygies'. Note that our method could be also applied to get similar results for more general categories (e.g. connected in codimension one algebraic sets).
A projective scheme $X$ is called `quadratic' if $X$ is scheme-theoretically cut out by homogeneous equations of degree 2. Furthermore, we say $X$ satisfies `property $\textbf{N}_{2,p}$' if it is quadratic and the quadratic ideal has only linear syzygies up to first $p$-th steps. In the present paper, we compare the linear syzygies of the inner projections with those of $X$ and obtain a theorem on `embedded linear syzygies' as one of our main results. This is the natural projection-analogue of `restricting linear syzygies' in the linear section case, \cite{EGHP1}. As an immediate corollary, we show that the inner projections of $X$ satisfy property $\textbf{N}_{2,p-1}$ for any reduced scheme $X$ with property $\textbf{N}_{2,p}$. Moreover, we also obtain the neccessary lower bound $(\codim X)\cdot p -\frac{p(p-1)}{2}$, which is sharp, on the number of quadrics vanishing on $X$ in order to satisfy $\textbf{N}_{2,p}$ and show that the arithmetic depths of inner projections are equal to that of the quadratic scheme $X$. These results admit an interesting `syzygetic' rigidity theorem on property $\textbf{N}_{2,p}$ which leads the classifications of extremal and next to extremal cases. For these results we develope the elimination mapping cone theorem for infinitely generated graded modules and improve the partial elimination ideal theory initiated by M. Green. This new method allows us to treat a wider class of projective schemes which can not be covered by the Koszul cohomology techniques, because these are not projectively normal in general.
Let $X\subset \P^N$ be a nondegenerate irreducible closed subvariety of dimension $n$ over the field of complex numbers and let $SX\subset\P^N$ be its secant variety. $X\subset\P^N$ is called `secant defective' if $\dim(SX)$ is strictly less than the expected dimension $2n+1$. In \cite{Z1}, F.L. Zak showed that for a secant defective manifold necessarily $N\le{n+2 \choose n}-1$ and that the Veronese variety $v_2(\P^n)$ is the only boundary case. Recently R. Mu$\tilde{\textrm{n}}$oz, J. C. Sierra, and L. E. Sol\'a Conde classified secant defective varieties next to this extremal case in \cite{MSS}. In this paper, we will consider secant defective manifolds $X\subset\P^N$ of dimension $n$ with $N={n+2 \choose n}-1-\epsilon$ for $\epsilon\ge0$. First, we will prove that $X$ is a $LQEL$-manifold of type $\delta=1$ for $\epsilon\le n-2$ (see Theorem \ref{main_thm}) by showing that the tangential behavior of $X$ is good enough to apply Scorza lemma. Then we will completely describe the above manifolds by using the classification of conic-connected manifolds given in \cite{IR1}. Our method generalizes previous results in \cite{Z1,MSS}.