negative kodaira dimension
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Author(s):  
Gianfranco Casnati

AbstractLet S be a regular surface endowed with a very ample line bundle $$\mathcal O_S(h_S)$$ O S ( h S ) . Taking inspiration from a very recent result by D. Faenzi on K3 surfaces, we prove that if $${\mathcal O}_S(h_S)$$ O S ( h S ) satisfies a short list of technical conditions, then such a polarized surface supports special Ulrich bundles of rank 2. As applications, we deal with general embeddings of regular surfaces, pluricanonically embedded regular surfaces and some properly elliptic surfaces of low degree in $$\mathbb {P}^{N}$$ P N . Finally, we also discuss about the size of the families of Ulrich bundles on S and we inspect the existence of special Ulrich bundles on surfaces of low degree.


2021 ◽  
Vol 9 ◽  
Author(s):  
Daniele Agostini ◽  
Ignacio Barros

Abstract We study smoothing of pencils of curves on surfaces with normal crossings. As a consequence we show that the canonical divisor of $\overline {\mathcal {M}}_{g,n}$ is not pseudoeffective in some range, implying that $\overline {\mathcal {M}}_{12,6}$ , $\overline {\mathcal {M}}_{12,7}$ , $\overline {\mathcal {M}}_{13,4}$ and $\overline {\mathcal {M}}_{14,3}$ are uniruled. We provide upper bounds for the Kodaira dimension of $\overline {\mathcal {M}}_{12,8}$ and $\overline {\mathcal {M}}_{16}$ . We also show that the moduli space of $(4g+5)$ -pointed hyperelliptic curves $\overline {\mathcal {H}}_{g,4g+5}$ is uniruled. Together with a recent result of Schwarz, this concludes the classification of moduli of pointed hyperelliptic curves with negative Kodaira dimension.


2020 ◽  
Vol 2020 (769) ◽  
pp. 55-86
Author(s):  
Kazuhiro Ito ◽  
Tetsushi Ito ◽  
Christian Liedtke

AbstractWe study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic p is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly less than {\frac{1}{2}(p-1)} (resp. p), then the surface has negative Kodaira dimension. We also prove similar, but weaker results hold for higher-dimensional varieties. Moreover, we show by example that our result is in some sense optimal. On our way, we obtain a sufficient criterion in terms of Jacobian numbers for the normalization of a curve over an imperfect field to be smooth.


2019 ◽  
Vol 62 (1) ◽  
pp. 123-135 ◽  
Author(s):  
ROBERTO LAFACE ◽  
PIOTR POKORA

AbstractWe give a bound on the H-constants of configurations of smooth curves having transversal intersection points only on an algebraic surface of non-negative Kodaira dimension. We also study in detail configurations of lines on smooth complete intersections $X \subset \mathbb{P}_{\mathbb{C}}^{n + 2}$ of multi-degree d = (d1, …, dn), and we provide a sharp and uniform bound on their H-constants, which only depends on d.


2017 ◽  
Vol 69 (1) ◽  
pp. 220-240 ◽  
Author(s):  
Tao Zheng

AbstractWe study the Chern-Ricci flow, an evolution equation of Hermitian metrics, on a family of Oeljeklaus–Toma (OT-) manifolds that are non-Kähler compact complex manifolds with negative Kodaira dimension. We prove that after an initial conformal change, the flow converges in the Gromov–Hausdorff sense to a torus with a flat Riemannianmetric determined by the OT-manifolds themselves.


2010 ◽  
Vol 146 (4) ◽  
pp. 919-928 ◽  
Author(s):  
Caucher Birkar

AbstractIn this paper, we prove that the log minimal model program in dimension d−1 implies the existence of log minimal models for effective lc pairs (e.g. of non-negative Kodaira dimension) in dimension d. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension d−1. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and of the Shokurov reduction theorem.


2006 ◽  
Vol 15 (3) ◽  
pp. 551-561 ◽  
Author(s):  
Jun-Muk Hwang ◽  
Stefan Kebekus ◽  
Thomas Peternell

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