Related papers: A result on resolutions of Veronese embeddings

This paper deals with syzygies of Segre embeddings. Let d >=3 and n_1, ..., n_d nonzero natural numbers. We prove that O(1,...., 1) on the product of P^{n_1}, ...,P^{n_d} satisfies Property N_p if and only if p <= 3.

We prove that the Veronese embedding of P^n of degree d with n\ge 2, d\ge 3 does not satisfy property N_p (according to Green and Lazarsfeld) if p\ge 3d-2. We conjecture that the converse holds. This is true for n=2 and for n=d=3.

The main result of this paper is the following: let d be a natural number >= 3; the line bundle O(1,...,1) on the product of d copies of the projective line P^1 satisfies Property N_p of Green-Lazarsfeld if and only if p <= 3.

This paper investigates property QR(3) for Veronese embeddings over an algebraically closed field of characteristic $3$. We determine the rank index of $(\mathbb{P}^n , \mathcal{O}_{\mathbb{P}^n} (d))$ for all $n \geq 2$, $d \geq 3$,…

The rational homology groups of the matching complexes are closely related to the syzygies of the Veronese embeddings. In this paper we will prove the vanishing of certain rational homology groups of matching complexes, thus proving that…

Let $C$ be a smooth (irreducible) curve of degree $d$ in $\mathbb{P}^{2}$. Let $\mathbb{P}^{2} \hookrightarrow \mathbb{P}^{5}$ be the Veronese embedding and let $\mathcal{I}_{C}$ denote the homogeneous ideal of $C$ on $\mathbb{P}^{5}$. In…

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…

We classify all the embeddings of $\mathbb{P}_n$ in a Grassmannian $Gr(1,N)$ such that the composition with Pl\"{u}cker embedding is given by a linear system of cubics on $\mathbb{P}_n$. As a corollary in the direction of the Hartshorne…

Let $X \subset \P^r$ be a nondegenerate projective variety and let $\nu_{\ell} : \P^r \to \P^N$ be the $\ell$-th Veronese embedding. In this paper we study the higher normality, defining equations and syzygies among them for the projective…

In this work we explicitly calculate syzygies of quadratic Veronese embedding $\mathbb{P}(V)\subset\mathbb{P}(\operatorname{Sym}^2V)$ as representations of the group $\operatorname{GL}(V)$. Also resolutions of the sheaves…

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…

We classify all the embeddings of P^n in a Grassmannian of lines G(1,N) such that the composition with Pl\"ucker is given by a linear system of quadrics of P^n.

For a vector bundle $\mathcal{E}$ of rank $n+1$ over a smooth projective curve $C$ of genus $g$, let $X=\P_C (\mathcal{E})$ with projection map $\pi:X\to C$. In this paper we investigate the minimal free resolution of homogeneous coordinate…

We describe the syzygy spaces for the Segre embedding $\mathbb{P}(U)\times\mathbb{P}(V)\subset\mathbb{P}(U\otimes V)$ in terms of representations of ${\rm GL}(U)\times {\rm GL}(V)$ and construct the minimal resolutions of the sheaves…

We exhibit an example of a line bundle M on a smooth complex projective variety Y s.t. M satisfy Property N_p for some p, the p-module of a minimal resolution of the ideal of the embedding of Y by M is nonzero and M^2 does not satisfy…

The quadratic Veronese embedding $\rho$ maps the point set $P$ of $\PG{n,F)$ into the point set of $PG({n+2 \choose 2}-1, F$ ($F$ a commutative field) and has the following well-known property: If $M\subset P$, then the intersection of all…

Let K be a valued field of characteristic p>0 with non-p-divisible value group. We show that every finite embedding problem for K whose kernel is a p-group is properly solvable.

This paper gives an explicit construction of the Tate resolution of sheaves arising from the d-fold Veronese embedding of P^n. Our description involves the Bezoutian of n+1 homogenous forms of degree d in n+1 variables. We give applications…

We construct a nonminimal graded free resolution of Segre embeddings of $P^1\times P^1$, although we don't compute all maps. We use this to prove an explicit formula for certain nonzero entries in the graded Betti table, at the end of the…

Which finite sets $P \subseteq \mathbb{Z}^r$ with $|P| \ge 3$ have the following property: for every $A \subseteq [N]^r$, there is some nonzero integer $d$ such that $A$ contains $(\alpha^{|P|} - o(1))N^r$ translates of $d \cdot P = \{d p :…