Let $X$ be the blowup of $\mathbb{P}^3$ at eight very general points. We give a complete description of its nef and effective cones. Moreover, we show that there exists a rational polyhedral fundamental domain for the action of a certain Weyl group on the effective movable cone of $X$.
We compute the facets of the effective cones of divisors on the blow-up of P^3 in up to five lines in general position. We prove that up to six lines these threefolds are weak Fano and hence Mori Dream Spaces.
We study $l$-very ample, ample and semi-ample divisors on the blown-up projective space $\mathbb{P}^n$ in a collection of points in general position. We establish Fujita's conjectures for all ample divisors with the number of points bounded above by $2n$ and for an infinite family of ample divisors with an arbitrary number of points.
We construct log resolutions of pairs on the blow-up of the projective space in an arbitrary number of general points and we discuss the semi-ampleness of the strict transforms. As an application we prove that the abundance conjecture holds for an infinite family of such pairs. For $n+2$ points, these strict transforms are F-nef divisors on the moduli space $\overline{\mathcal{M}}_{0,n+3}$ in a Kapranov's model: we show that all of them are nef.
The objective is to show the construction of an Ulrich vector bundle on the blowing-up $\widetilde X$ of a nonsingular projective variety $X$ at a closed point, where the original variety is embedded by a very ample divisor $H$ and carries an Ulrich vector bundle. In order to achieve this result, we aim to find a suitable very ample divisor on $\widetilde X$, which is dependent on $H$. At the end, we take into consideration some applications to surfaces with regards to minimal models and their Kodaira dimension.
In this paper, we prove that $\mathbb{P}^2$ blown up at seven general points admits a conic bundle structure over $\mathbb{P}^1$ and it can be embedded as $(2,2)$ divisor in $\mathbb{P}^{1}\times\mathbb{P}^{2}$. Conversely, any smooth surface in the complete linear system $\mid (2,2) \mid$ of $\mathbb{P}^{1}\times\mathbb{P}^{2}$ is obtained by blowing up $\mathbb{P}^2$ at seven points. We also show any smooth surface linearly equivalent to $(2,2)$ in $\mathbb{P}^{1}\times\mathbb{P}^{2}$ has at most four $(-2)$ curves (the curve which has self intersection $(-2)$).
We study divisors in the blow-up of $\mathbb{P}^n$ at points in general position that are non-special with respect to the notion of linear speciality introduced in [5]. We describe the cohomology groups of their strict transforms via the blow-up of the space along their linear base locus. We extend the result to non-effective divisors that sit in a small region outside the effective cone. As an application, we describe linear systems of divisors in $\mathbb{P}^n$ blown-up at points in star configuration and their strict transforms via the blow-up of the linear base locus.
This article is motivated by the authors interest in the geometry of the Mori dream space $\mathbb{P}^4$ blown up in $8$ general points. In this article we develop the necessary technique for determining Weyl orbits of linear cycles for the four-dimensional case, by explicit computations in the Chow ring of the resolution of the standard Cremona transformation. In particular, we close this paper with applications to the question of the dimension of the space global sections of effective divisors having at most $8$ base points.
The geometry of eight points in projective space: Representation theory, Lie theory, dualities
本文讨论了P^1中8点空间（GIT商）M_8的几何，以及P^3中8点GIT商的Gale-quotient N '_8。空间M_8在P^{13}中有一个自然的嵌入，或者更准确地说，是S_8-表示V_{4,4}的投影化。在P^{13}中存在一个S_8-斜三次C。M_8落在斜三次曲线C上的事实是M_8关于超椭圆曲线公式的一个推论，但更确切的是：M_8是C的奇异轨迹。这些构造产生了M_8的自由分解，并且通常用于确定截断P^1中n个点的GIT商的“单一”方程。空间N '_8在P^{13}中具有自然嵌入，或者更精确地，PV_{2,2,2,2}。存在一个含有N '_8的斜五次曲线Q，N'_8是斜五次曲线Q的奇异轨迹.斜三次C和斜五次Q是射影对偶的。(In特别地，它们是令人惊讶的奇异的，在具有非常低程度的对偶的意义上。对偶映射下的斜三次曲面上的因子是割线簇Sec（M_8），收缩Sec（M_8）- - > N '_8通过二次曲面上的8个点的空间因子化N_8。我们猜想被对偶映射吹倒的斜五次曲线上的因子是N '_8（四割线并的闭包）的四割线簇，而五次曲线Q是三割线簇.由此产生的图片扩展了经典的对偶性在6点的情况下之间的塞格雷三次三倍和Igusa四次三倍。我们注意到，有许多几何上的自然变种是（相关）显着奇异三次超曲面的奇异轨迹。本文的部分内容出现在arXiv/0809.1233中。
Let $(S,L)$ be a polarized K3 surface with $\mathrm{Pic}(S) = \mathbb{Z}[L]$ and $L\cdot L=2g-2$, let $C$ be a nonsingular curve of genus $g-1$ and let $f:C\to S$ be such that $f(C) \in \vert L \vert$. We prove that the Gaussian map $Φ_{ω_C(-T)}$ is non-surjective, where $T$ is the degree two divisor over the singular point $x$ of $f(C)$. This generalizes a result of Kemeny with an entirely different proof. It uses the very ampleness of $C$ on the blown-up surface $\widetilde S$ of $S$ at $x$ and a theorem of L'vovski.
In this paper, we classify relatively minimal genus-$1$ holomorphic Lefschetz pencils up to smooth isomorphism. We first show that such a pencil is isomorphic to either the pencil on $\mathbb{P}^1\times \mathbb{P}^1$ of bi-degree $(2,2)$ or a blow-up of the pencil on $\mathbb{P}^2$ of degree $3$, provided that no fiber of a pencil contains an embedded sphere. (Note that one can easily classify genus-$1$ Lefschetz pencils with an embedded sphere in a fiber.) We further determine the monodromy factorizations of these pencils and show that the isomorphism class of a blow-up of the pencil on $\mat…
