Higher-Dimensional Algebraic Geometry

By O. Debarre. 2001

345 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dca8b9c057aab82f1e2285da32a58594be1b1d4a

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In Algebra and Number Theory

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Citations in the corpus

The Density of Rational Points on Non‐Singular Hypersurfaces, I (2005, background): “3 contains at most ﬁnitely many such subvarieties. Let k be a ﬁeld. Let X ⊂ Pn k be a hypersurface of degree d > 1. For each integer m > 0, denote by Fm(X) the Fano scheme of m-planes in X, cf. [1]. There are a number of natural questions about Fm(X): is this scheme non-empty, is this scheme c”
Rationally connected foliations after Bogomolov and McQuillan (2005, background): “8 S. KEBEKUS, L. SOLA CONDE, AND M. TOMA Recall [ Deb01 , thm.”
Space of Ricci Flows I (2009, background): “sequently are Ka¨hler Einstein metric. Suppose LM ∈ K (n,A) and λ>0. Then it is clear that c1(M) >0, or M is Fano. Note that for every Fano manifold, we have a uniform bound cn 1(M) ≤ C(n)(c.f. [40]). This implies that 1 A ≤ Vol(M) = cn 1(L) = λ−ncn 1(M) ≤ Cλ−n. 119 So λis bounded away from ab”
Symplectic resolutions for nilpotent orbits (2002, background): “Proof. By an argument of [ Deb ] (p. 28), V being normal and locally Q-factorial implies that for any birational map π : Z → V , the exceptional set E ⊂ Z of π is of pure codimension 1 in Z. Now the corollary follows in the same way as the precedent proposition.”
A decomposition theorem for projective manifolds with nef anticanonical bundle (2017, methodology): “We use the terminology of [Deb01] for birational geometry and [Laz04a] for (algebraic) notions of positivity.”
Singular symplectic moduli spaces (2005, background): “This implies that the exceptional locus must be a divisor (see [7] no.”
Varieties with ample cotangent bundle (2003, methodology): “nvertible sheaf on A, we deﬁne, for any section s of L with divisor H, a section ∂s of L|H by the requirement that for any open set 2One can avoid this unpleasant argument by using the rigidity lemma [D3], Lemma 1.15(a) or the argument of [B1], Theorem 1. 5 U of A and any trivialization ϕ : OU −→∼ L”
Cohomological characterizations of projective spaces and hyperquadrics (2007, background): “For more details see [Kol96], [ Deb01 ], or [AK03].”