Anti-pluricanonical systems on Fano varieties
By C. Birkar. Annals of Mathematics, 2016
229 citations recorded by the source corpus.
In this paper, we study the linear systems $|-mK_X|$ on Fano varieties $X$ with klt singularities. In a given dimension $d$, we prove $|-mK_X|$ is non-empty and contains an element with "good singularities" for some natural number $m$ depending only on $d$; if in addition $X$ is $\epsilon$-lc for some $\epsilon>0$, then we show that we can choose $m$ depending only on $d$ and $\epsilon$ so that $|-mK_X|$ defines a birational map. Further, we prove Shokurov's conjecture on boundedness of complements, and show that certain classes of Fano varieties form bounded families.
Subfields: Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/ece822593c903e234820c6c7da060654aacaed38
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In Geometry and Topology
Citations in the corpus
- Singularities of linear systems and boundedness of Fano varieties (2016, background): “ped with a boundary ∆ such that KX + ∆ ≡ 0, (X,∆) is ǫ-lc, and such that the coefficients of ∆ belong to a DCC set [11], or more generally when the coefficients of ∆ are bounded from below away from zero [3], form a bounded family. Corollary 1.2. Let dbe a natural number and ǫa positive real number. The”
- ACC for Minimal Log Discrepancies of Exceptional Singularities (2019, unknown): “:3) is a finite set of rational numbers, Theorem 1.3 was proved for surfaces and threefolds [39, 40, 44], and more recently, by Birkar [3, Theorems 1.7, 1.8] in any dimension.”
- Geometric Manin’s conjecture and rational curves (2017, unknown): “roved this statement conditional on a version of the BorisovAlexeev-Borisov Conjecture. [HJ16, Theorem 1.1] noticed that when L is big and semiample one can apply the weaker boundedness statements of [Bir16a] and [DC16]. Since [Bir16b] has settled the Borisov-Alexeev-Borisov Conjecture, the semiampl”
- Boundedness of Q-Fano varieties with degrees and alpha-invariants bounded from below (2017, background): “ near x with a unique non-klt place whose center contains x, that center is G, • (X,∆) is not klt at y, and • either dimG = 0 or (A|G)dimG ≤ dd. 2.5. Birkar’s results. We recall several theorems from [Bir16a]. The following theorem provides a criterion of boundedness of certain Q-Fano varieties, whi”
- Uniqueness of K-polystable degenerations of Fano varieties (2018, result): “We list the main ones: (I) Boundedness: There is a positive integer N = N(n,V) such that if X ∈ MKss n,V(k), then −NKX is a very ample Cartier divisor. This is settled in [Jia17] using the results in [Bir16]. (II) Zariski openness: If X→ Sis a family of Q-Fano varieties, then the locus where the fibe”