Normality and quadraticity for special ample line bundles on toric varieties arising from root systems

Projective normality of abelian varieties

We show that ample line bundles L on a g-dimensional simple abelian variety, with h^0(L)> 2^g.g!, give projectively normal embeddings, for all g>0.

Projective Normality Of Algebraic Curves And Its Application To Surfaces

Let $L$ be a very ample line bundle on a smooth curve $C$ of genus $g$ with $\frac{3g+3}{2}<°L\le 2g-5$. Then $L$ is normally generated if $°L>\max\{2g+2-4h^1(C,L), 2g-\frac{g-1}{6}-2h^1(C,L)\}$. Let $C$ be a triple covering of genus $p$ curve $C'$ with $C\stackrelϕ\to C'$ and $D$ a divisor on $C'$ with $4p<°D< \frac{g-1}{6}-2p$. Then $K_C(-ϕ^*D)$ becomes a very ample line bundle which is normally generated. As an application, we characterize some smooth projective surfaces.

Projective normality of abelian surfaces of type (1,2d)

We show that an abelian surface embedded in P^N by a very ample line bundle L of type (1,2d) is projectively normal if and only if d>=4. This completes the study of the projective normality of abelian surfaces embedded by complete linear systems.

Projective normality of Abelian threefolds

In this article, we show that very ample line bundles of type (1,1,d) on an abelian threefold A, with NS(A)\simeq \ze, give projectively normal embeddings, if d=22,26 or when d\geq 28.

On the k-normality of some projective manifolds

A long standing conjecture, known to us as the Eisenbud Goto conjecture, states that an n-dimensional variety embedded with degree $d$ in the $N$- dimensional projective space is $(d-(N-n)+1)$-regular in the sense of Castelnuovo-Mumford. In this work the conjecture is proved for all smooth varieties $X$ embedded by the complete linear system associated with a very ample line bundle $L$ such that $Δ(X,L) \le 5$ where $Δ(X,L) = \dim{X} + °{X} -h^0(L).$ As a by-product of the proof of the above result the projective normality of a class of surfaces of degree nine in $\Pin{5}$ which was left as an…

An Eisenbud-Goto-type Upper Bound for the Castelnuovo-Mumford Regularity of Fake Weighted Projective Spaces

We will give an upper bound for the $k$-normality of very ample lattice simplices, and then give an Eisenbud-Goto-type bound for some special classes of projective toric varieties.

Projective normality of canonical symmetric squares

Recall that a smooth complex projective curve has a very ample canonical bundle when it is non-hyperelliptic, and according to a theorem of M. Noether the resulting embedding is projectively normal. A theorem of Petri further asserts that the homogeneous ideal is generated by quadrics if the curve is neither trigonal nor a smooth plane quintic. In this note, we prove an analogue of Noether's theorem for the symmetric square of the curve - namely, the canonical bundle of the symmetric square determines a projectively normal embedding exactly when the curve itself is neither hyperelliptic, trigo…

加权射影空间上的极丰富线丛与加权吹胀

Very ample line bundles on weighted projective spaces and weighted blowups

该论文研究加权射影空间上的线丛丰富性问题，给出极丰富性的尖锐判据，证明权重两两互素时$\boldsymbol{\textit{O}}(d)$极丰富，还探讨了加权吹胀等相关代数几何内容。

On torus quotients of Schubert varieties in orthogonal Grassmannian-II

Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n+2}$. In this article, we prove projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle.

Syzygies of surfaces of general type

We prove new results on projective normality, normal presentation and higher syzygies for a surface of general type $X$ embedded by adjoint line bundles $L_r = K + rB$, where $B$ is a base point free, ample line bundle. Our main results determine the $r$ for which $L_r$ has $N_p$ property. In corollaries, we will relate the bounds on $r$ to the regularity of $B$. Examples in the last section show that several results are optimal.

Number variance of random zeros

The main results of this article are asymptotic formulas for the variance of the number of zeros of a Gaussian random polynomial of degree $N$ in an open set $U \subset C$ as the degree $N \to \infty$, and more generally for the zeros of random holomorphic sections of high powers of any positive line bundle over any Riemann surface. The formulas were conjectured in special cases by Forrester and Honner. In higher dimensions, we give similar formulas for the variance of the volume inside a domain $U$ of the zero hypersurface of a random holomorphic section of a high power of a positive line bun…