scholarly journals On Mori chamber and stable base locus decompositions

2019 ◽  
Vol 373 (3) ◽  
pp. 1667-1700 ◽  
Author(s):  
Antonio Laface ◽  
Alex Massarenti ◽  
Rick Rischter
Keyword(s):  
Base Locus

scholarly journals The diminished base locus is not always closed

2014 ◽  
Vol 150 (10) ◽  
pp. 1729-1741 ◽  
Author(s):  
John Lesieutre
Keyword(s):  
Blow Up ◽  
Weak Sense ◽  
Prime Divisors ◽  
General Fiber ◽  
Base Locus ◽  
The Family

AbstractWe exhibit a pseudoeffective $\mathbb{R}$-divisor ${D}_{\lambda }$ on the blow-up of ${\mathbb{P}}^{3}$ at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus ${\boldsymbol{B}}_{-}({D}_{\lambda })={\bigcup }_{A\,\text{ample}}\boldsymbol{B}({D}_{\lambda }+A)$ is not closed and that ${D}_{\lambda }$ does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an $\mathbb{R}$-divisor on the family of blow-ups of ${\mathbb{P}}^{2}$ at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.

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 Seshadri Constants for Curve Classes

2019 ◽  
Author(s):  
Mihai Fulger
Keyword(s):  
Normal Bundle ◽  
Projective Variety ◽  
Smooth Curve ◽  
Smooth Projective Variety ◽  
Projective Varieties ◽  
Seshadri Constants ◽  
Base Locus ◽  
Nef Divisors

Abstract We develop a local positivity theory for movable curves on projective varieties similar to the classical Seshadri constants of nef divisors. We give analogues of the Seshadri ampleness criterion, of a characterization of the augmented base locus of a big and nef divisor, and of the interpretation of Seshadri constants as an asymptotic measure of jet separation. As application, we show in any characteristic that if $C$ is a smooth curve with ample normal bundle in a smooth projective variety then the class of $C$ is in the strict interior of the Mori cone. This was conjectured by Peternell and proved by Ottem and Lau in Characteristic 0.

scholarly journals Special cubic Cremona transformations of ℙ6 and ℙ7

2019 ◽  
Vol 19 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Giovanni Staglianò
Keyword(s):  
Cremona Transformation ◽  
Cremona Transformations ◽  
Base Locus ◽  
Famous Result

Abstract A famous result of Crauder and Katz (1989) concerns the classification of special Cremona transformations whose base locus has dimension at most two. They also proved that a special Cremona transformation with base locus of dimension three has to be one of the following: 1) a quinto-quintic transformation of ℙ5; 2) a cubo-quintic transformation of ℙ6; or 3) a quadro-quintic transformation of ℙ8. Special Cremona transformations as in Case 1) have been classified by Ein and Shepherd-Barron (1989), while in our previous work (2013), we classified special quadro-quintic Cremona transformations of ℙ8. Here we consider the problem of classifying special cubo-quintic Cremona transformations of ℙ6, concluding the classification of special Cremona transformations whose base locus has dimension three.

scholarly journals RANK FOUR SYMPLECTIC BUNDLES WITHOUT THETA DIVISORS OVER A CURVE OF GENUS TWO

2008 ◽  
Vol 19 (04) ◽  
pp. 387-420 ◽  
Author(s):  
GEORGE H. HITCHING
Keyword(s):  
Line Bundle ◽  
Moduli Space ◽  
Vector Bundles ◽  
Projective Variety ◽  
Theta Divisor ◽  
Weierstrass Points ◽  
Base Locus ◽  
Canonical Bijection ◽  
Genus Two ◽  
Irreducible Projective Variety

The moduli space [Formula: see text] of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree -1 over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Raynaud and Beauville using Fourier–Mukai transforms. As an application, we prove that the map sending a symplectic vector bundle to its theta divisor is a surjective map from [Formula: see text] to the space of even 4Θ divisors on the Jacobian variety of the curve.

Grothendieck–Lefschetz theorem with base locus

2016 ◽  
Vol 212 (1) ◽  
pp. 107-122 ◽  
Author(s):  
John Brevik ◽  
Scott Nollet
Keyword(s):  
Base Locus ◽  
Lefschetz Theorem

Quadric Hypersurfaces Containing a Projectively Normal Curve

2005 ◽  
Vol 12 (2) ◽  
pp. 201-206
Author(s):  
Edoardo Ballico ◽  
Changho Keem
Keyword(s):  
Line Bundles ◽  
Normal Curve ◽  
Connected Component ◽  
Base Locus

Abstract Let 𝐶 ⊂ 𝐏𝑛 be a smooth projectively normal curve. Let 𝑍 be the scheme-theoretic base locus of 𝐻0(𝐏𝑛, 𝐼𝐶(2)) and 𝑍′ the connected component of 𝑍 containing 𝐶. Here we show that 𝑍′ = 𝐶 in certain cases (e.g., non-special line bundles with degree near to 2𝑝𝑎 (𝐶)–2 or certain special line bundles on general 𝑘-gonal curves).

scholarly journals The augmented base locus of real divisors over arbitrary fields

2016 ◽  
Vol 368 (3-4) ◽  
pp. 905-921 ◽  
Author(s):  
Caucher Birkar
Keyword(s):  
Base Locus

scholarly journals Stable base locus decompositions of Kontsevich moduli spaces

2010 ◽  
Vol 59 (2) ◽  
pp. 435-466 ◽  
Author(s):  
Dawei Chen ◽  
Izzet Coskun
Keyword(s):  
Moduli Spaces ◽  
Base Locus
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