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.

scholarly journals LOCAL VANISHING AND HODGE FILTRATION FOR RATIONAL SINGULARITIES

2018 ◽  
Vol 19 (3) ◽  
pp. 801-819
Author(s):  
Mircea Mustaţă ◽  
Sebastián Olano ◽  
Mihnea Popa
Keyword(s):  
Toric Variety ◽  
Exceptional Divisor ◽  
Resolution Of Singularities ◽  
Isolated Singularities ◽  
Rational Singularities ◽  
Smooth Complex ◽  
Hodge Filtration ◽  
Generation Level ◽  
Image Position

Given an $n$-dimensional variety $Z$ with rational singularities, we conjecture that if $f:Y\rightarrow Z$ is a resolution of singularities whose reduced exceptional divisor $E$ has simple normal crossings, then $$\begin{eqnarray}\displaystyle R^{n-1}f_{\ast }\unicode[STIX]{x1D6FA}_{Y}(\log E)=0. & & \displaystyle \nonumber\end{eqnarray}$$ We prove this when $Z$ has isolated singularities and when it is a toric variety. We deduce that for a divisor $D$ with isolated rational singularities on a smooth complex $n$-dimensional variety $X$, the generation level of Saito’s Hodge filtration on the localization $\mathscr{O}_{X}(\ast D)$ is at most $n-3$.

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.

The sheaf of relative differentials of a fibred surface

1993 ◽  
Vol 114 (3) ◽  
pp. 461-470
Author(s):  
Fernando Serrano
Keyword(s):  
Linear Systems ◽  
Smooth Curve ◽  
Base Point ◽  
Projective Surface ◽  
Positive Part ◽  
Rational Singularities ◽  
Smooth Projective Surface ◽  
Genus 2

AbstractLet Φ: S → C denote a fibration from a smooth projective surface onto a smooth curve, with fibres of genus ≥2. The double dual of the sheaf of relative differentials has been studied by F. Serrano [14]. There, it was proved that dim grows asymptotically as the square of n in case Φ is not isotrivial (i.e. fibres vary in modulus), and the converse holds true in most cases, in a way that can be made precise. In the non-isotrivial case, the present paper provides further information about by analysing the linear systems for large n. If P denotes the positive part of in its Zariski decomposition, then it is shown that |rP| is eventually base-point free for some r > 0. Furthermore, Proj is a normal projective surface, fibred over C, birational to S, and with only rational singularities.

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.

Uniruledness of stable base loci of adjoint linear systems via Mori theory

2013 ◽  
Vol 275 (1-2) ◽  
pp. 499-507 ◽  
Author(s):  
Sébastien Boucksom ◽  
Amaël Broustet ◽  
Gianluca Pacienza
Keyword(s):  
Linear Systems ◽  
Mori Theory ◽  
Base Loci

Very Ample Linear Systems on Blowings-Up at General Points of Projective Spaces

2002 ◽  
Vol 45 (3) ◽  
pp. 349-354 ◽  
Author(s):  
Marc Coppens
Keyword(s):  
Projective Space ◽  
Linear Systems ◽  
Exceptional Divisor ◽  
Closed Field ◽  
Invertible Sheaf ◽  
Algebraically Closed Field ◽  
Structure Sheaf ◽  
Blowing Up ◽  
Dimensional Projective Space ◽  
Ample Invertible Sheaf

AbstractLet Pn be the n-dimensional projective space over some algebraically closed field k of characteristic 0. For an integer t ≥ 3 consider the invertible sheaf O(t) on Pn (Serre twist of the structure sheaf). Let , the dimension of the space of global sections of O(t), and let k be an integer satisfying 0 < k ≤ N − (2n + 2). Let P1,…,Pk be general points on Pn and let π : X → Pn be the blowing-up of Pn at those points. Let Ei = π−1(Pi) with 1 ≤ i ≤ k be the exceptional divisor. Then M = π*(O(t)) ⊗ OX(−E1 — … — Ek) is a very ample invertible sheaf on X.

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