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

Abstract: We show that the border rank of the 4 × 4 determinant tensor is at least 12 over 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.

Abstract: We generalize the celebrated Fröberg's theorem to embedded joins of copies of a simplicial complex, namely higher secant complexes to the simplicial complex, in terms of property Nq+1,p due to Green and Lazarsfeld. Furthermore, we investigate combinatorial phenomena parallel to geometric ones observed for higher secant varieties of minimal degree.

Abstract: 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 R(detₙ) completely separate the determinant and the permanent tensors by their tensor ranks. Furthermore, we determine the exact tensor ranks R(det₄)=12 and R(perm₄)=8 over arbitrary field of characteristic ≠ 2.

Abstract: 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.

Abstract: 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., $\deg(\sigma_4(v_3(\mathbb{P}^3))) = 105$ 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.