scholarly journals Remarks on Arithmetic Restricted Volumes and Arithmetic Base Loci

2016 ◽  
Vol 52 (4) ◽  
pp. 435-495
Author(s):  
Hideaki Ikoma
Keyword(s):  
Base Loci

scholarly journals On the non-divisorial base locus of big and nef line bundles on K3[2]-type varieties

2020 ◽  
pp. 1-27
Author(s):  
Ulrike Rieß
Keyword(s):  
Moduli Spaces ◽  
Hilbert Scheme ◽  
Ample Line Bundle ◽  
Base Point ◽  
K3 Surfaces ◽  
Line Bundles ◽  
Codimension Two ◽  
Base Locus ◽  
Base Loci ◽  
Symplectic Varieties

Abstract We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized K3[2]-type varieties, the polarization is base point free.

scholarly journals On base loci of higher fundamental forms of toric varieties

2020 ◽  
Vol 224 (12) ◽  
pp. 106447
Author(s):  
Antonio Laface ◽  
Luca Ugaglia
Keyword(s):  
Toric Varieties ◽  
Base Loci

scholarly journals Augmented base loci and restricted volumes on normal varieties, II: The case of real divisors

2015 ◽  
Vol 159 (3) ◽  
pp. 517-527
Author(s):  
ANGELO FELICE LOPEZ
Keyword(s):  
Open Subset ◽  
Projective Variety ◽  
Closed Field ◽  
Projective Varieties ◽  
Algebraically Closed Field ◽  
Smooth Complex ◽  
Normal Varieties ◽  
Restricted Volume ◽  
Complex Projective ◽  
Base Loci

AbstractLet X be a normal projective variety defined over an algebraically closed field and let Z be a subvariety. Let D be an ${\mathbb R}$-Cartier ${\mathbb R}$-divisor on X. Given an expression (*) D$\sim_{\mathbb R}$t1H1 +. . .+ tsHs with ti ∈ ${\mathbb R}$ and Hi very ample, we define the (*)-restricted volume of D to Z and we show that it coincides with the usual restricted volume when Z$\not\subseteq$B+(D). Then, using some recent results of Birkar [Bir], we generalise to ${\mathbb R}$-divisors the two main results of [BCL]: The first, proved for smooth complex projective varieties by Ein, Lazarsfeld, Mustaţă, Nakamaye and Popa, is the characterisation of B+(D) as the union of subvarieties on which the (*)-restricted volume vanishes; the second is that X − B+(D) is the largest open subset on which the Kodaira map defined by large and divisible (*)-multiples of D is an isomorphism.

scholarly journals Zariski chambers, volumes, and stable base loci

2004 ◽  
Vol 2004 (576) ◽  
Author(s):  
Th. Bauer ◽  
A. Küronya ◽  
T. Szemberg
Keyword(s):  
Base Loci

scholarly journals Log canonical pairs with good augmented base loci

2014 ◽  
Vol 150 (4) ◽  
pp. 579-592 ◽  
Author(s):  
Caucher Birkar ◽  
Zhengyu Hu
Keyword(s):  
Minimal Model ◽  
Surjective Morphism ◽  
Normal Variety ◽  
Base Locus ◽  
Log Canonical ◽  
Canonical Pair ◽  
Interesting Special Case ◽  
Base Loci ◽  
Special Case

AbstractLet $(X,B)$ be a projective log canonical pair such that $B$ is a $\mathbb{Q}$-divisor, and that there is a surjective morphism $f: X\to Z$ onto a normal variety $Z$ satisfying $K_X+B\sim _{\mathbb{Q}} f^*M$ for some big $\mathbb{Q}$-divisor $M$, and the augmented base locus ${\mathbf{B}}_+(M)$ does not contain the image of any log canonical centre of $(X,B)$. We will show that $(X,B)$ has a good log minimal model. An interesting special case is when $f$ is the identity morphism.

scholarly journals Asymptotic invariants of base loci

2006 ◽  
Vol 56 (6) ◽  
pp. 1701-1734 ◽  
Author(s):  
Lawrence Ein ◽  
Robert Lazarsfeld ◽  
Mircea Mustaţă ◽  
Michael Nakamaye ◽  
Mihnea Popa
Keyword(s):  
Asymptotic Invariants ◽  
Base Loci

scholarly journals Asymptotic base loci via Okounkov bodies

2018 ◽  
Vol 323 ◽  
pp. 784-810 ◽  
Author(s):  
Sung Rak Choi ◽  
Yoonsuk Hyun ◽  
Jinhyung Park ◽  
Joonyeong Won
Keyword(s):  
Base Loci

scholarly journals Homological Projective Duality for Linear Systems with Base Locus

2018 ◽  
Vol 2020 (21) ◽  
pp. 7829-7856 ◽  
Author(s):  
Francesca Carocci ◽  
Zak Turčinović
Keyword(s):  
Linear Systems ◽  
Dual Pair ◽  
Resolution Of Singularities ◽  
Smooth Variety ◽  
Rational Singularities ◽  
Base Locus ◽  
Projective Duality ◽  
Blowing Up ◽  
Linear Sections ◽  
Base Loci

Abstract We show how blowing up varieties in base loci of linear systems gives a procedure for creating new homological projective duals from old. Starting with a homological projective (HP) dual pair $X,Y$ and smooth orthogonal linear sections $X_L,Y_L$, we prove that the blowup of $X$ in $X_L$ is naturally HP dual to $Y_L$. The result also holds true when $Y$ is a noncommutative variety or just a category. We extend the result to the case where the base locus $X_L$ is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a weakly crepant categorical resolution of singularities of $Y_L$. Finally we give examples where, starting with a noncommutative $Y$, the above process nevertheless gives geometric HP duals.

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