Nef Cone of a Generalized Kummer 4-fold

In this article, we prove a finiteness result on the number of log minimal models for 3-folds in $\operatorname{char}p>5$. We then use this result to prove a version of Batyrev’s conjecture on the structure of nef cone of curves on 3-folds in characteristic $p>5$. We also give a proof of the same conjecture in full generality in characteristic 0. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor $K_{X}$ of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on $X$.

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Nef cone volumes and discriminants of abelian surfaces

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2017.09.016 ◽

2018 ◽

Vol 222 (8) ◽

pp. 2330-2341

Author(s):

Thomas Bauer ◽

Carsten Bornträger

Keyword(s):

Abelian Surfaces ◽

Nef Cone

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The Nef Cone of the Hilbert Scheme of Points on Rational Elliptic Surfaces and the Cone Conjecture

Canadian Mathematical Bulletin ◽

10.4153/s0008439520000387 ◽

2020 ◽

pp. 1-12

Author(s):

John Kopper

Keyword(s):

Hilbert Scheme ◽

Elliptic Surface ◽

Elliptic Surfaces ◽

Hilbert Scheme Of Points ◽

Rational Elliptic Surfaces ◽

Nef Cone ◽

Rational Elliptic Surface

Abstract We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison–Kawamata cone conjecture holds for these nef cones.

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Toric varieties and Gröbner bases: the complete $$\mathbb {Q}$$-factorial case

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-020-00452-w ◽

2020 ◽

Vol 31 (5-6) ◽

pp. 461-482

Author(s):

Michele Rossi ◽

Lea Terracini

Keyword(s):

Toric Varieties ◽

Toric Ideals ◽

Projective Embedding ◽

Complete Case ◽

Open Problems ◽

Toric Ideal ◽

Definition Of ◽

Nef Cone ◽

Special Fibre

Abstract We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors V as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it computes all complete and simplicial fans supported by V and lead us to formulate a topological-combinatoric conjecture about the definition of a fan. On the other hand, we adapt the Sturmfels’ arguments on the Gröbner fan of toric ideals to our complete case; we give a characterization of the Gröbner region and show an explicit correspondence between Gröbner cones and chambers of the secondary fan. A homogenization procedure of the toric ideal associated to V allows us to employing GFAN and related software in producing our second algorithm. The latter turns out to be much faster than the former, although it can compute only the projective fans supported by V. We provide examples and a list of open problems. In particular we give examples of rationally parametrized families of $$\mathbb {Q}$$ Q -factorial complete toric varieties behaving in opposite way with respect to the dimensional jump of the nef cone over a special fibre.

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