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

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.

Degeneration of Kummer surfaces

We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field K of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction of Kulikov models is closely related to an earlier construction of Künnemann, which produces semistable models of Abelian varieties. It is well known that the special fibre of a strict Kulikov model belongs to one of three types, and we shall prove that the type of the special fibre of a strict Kulikov model of a Kummer surface and the toric rank of a corresponding Abelian surface are determined by each other. We also study the relationship between this invariant and the Galois representation on the second ℓ-adic cohomology of the Kummer surface. Finally, we apply our results, together with earlier work of Halle–Nicaise, to give a proof of the monodromy conjecture for Kummer surfaces in equal characteristic zero.

DEFORMATION THEORY OF THE CHOW GROUP OF ZERO CYCLES

We study the deformations of the Chow group of zerocycles of the special fibre of a smooth scheme over a Henselian discrete valuation ring. Our main tools are Bloch’s formula and differential forms. As a corollary we get an algebraization theorem for thickened zero cycles previously obtained using idelic techniques. In the course of the proof we develop moving lemmata and Lefschetz theorems for cohomology groups with coefficients in differential forms.

Cycle classes in overconvergent rigid cohomology and a semistable Lefschetz theorem

In this paper we prove a semistable version of the variational Tate conjecture for divisors in crystalline cohomology, showing that for $k$ a perfect field of characteristic $p$ , a rational (logarithmic) line bundle on the special fibre of a semistable scheme over $k\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ lifts to the total space if and only if its first Chern class does. The proof is elementary, using standard properties of the logarithmic de Rham–Witt complex. As a corollary, we deduce similar algebraicity lifting results for cohomology classes on varieties over global function fields. Finally, we give a counter-example to show that the variational Tate conjecture for divisors cannot hold with $\mathbb{Q}_{p}$ -coefficients.

Prevention of concrete structures from collapsing

At the end of the 20th century requirements on using electrical properties of building materials emerged for application in heating of trafficable surfaces, grounding of electrostatic charges in floors, shielding of electro-magnetic fields and diagnosis of concrete structure state in the course of time. For this reason, it was necessary to design special fibre-cement elements able to transfer any mechanical impulse to an electricallymeasured signal detected as a change in electrical resistance with computer outputs. Regarding previous research studies, it was concluded that special fibre-cement composites are able to conduct electric current under specific conditions. This property is ensured by using of various kinds of carbon materials. Though carbon fibres are less conductive than metal fibres, composites with carbon fibres were evaluated as better current conductors than the composites with metal fibres. By means of various kinds of carbon particles and fibres it is possible to design cement composites with an ability to monitor changes in electrical conductivity of concretes. The designed composites are assembled with conductive wires and connected with a special electronic equipment for monitoring of changes in alternate voltage passing through the tensometer within mechanical loading of a concrete element in which the composite is integrated. The tensometers are placed preferably into parts of the concrete elements subjected to compression, such as simple reinforced columns or upper parts of longitudinal beams. Several tests of repeated loading and simultaneous monitoring of vertical as well as horizontal prefabricated concrete elements were carried out and evaluated.

The Influence of the Fastener Hole Preparation Method on the Fastener Pull-Through Process in a Carbon Composite

This article studies the pull-through resistance of a titanium carbon fibre-epoxy resin laminate fastener. Coupons with fastener holes made with different methods were compared – drilled, milled on a CNC plotter and special fibre application during laminate production. The tests were conducted according to the ASTM D7332 test standard. The studies showed that the fastener hole preparation method impacts the laminate’s resistance to fastener pull-through. Coupons with holes made with standard (drilling and milling) methods showed fastener pull-through resistance higher, on average, by 6.5% than in coupons with holes placed during plate production. Fastener work to rupture was also higher for coupons with milled and drilled holes. Microscopic observations in UV-light, using a fluorescent penetrant, showed differences in failure mechanisms between individual coupons, especially the lack of fibres in the 0° direction, in immediate vicinity to a hole prepared during laminate application.

Tate Cycles on Abelian Varieties with Complex Multiplication

We consider Tate cycles on an Abelian variety A defined over a sufficiently large number field K and having complexmultiplication. We show that there is an effective bound C = C(A, K) so that to check whether a given cohomology class is a Tate class on A, it suffices to check the action of Frobenius elements at primes v of norm ≤ C. We also show that for a set of primes v of K of density 1, the space of Tate cycles on the special fibre Av of the Néron model of A is isomorphic to the space of Tate cycles on A itself.

Even-Relative-Dimensional Vanishing Cycles in Bivariant Intersection Theory

For a smooth variety proper over a curve having a fibre with isolated ordinary quadratic singularities, it is well-known that we have the vanishing cycles associated to the singularities in the étale cohomology of the geometric generic fibre. The base-change by a double cover of the base curve ramified at the image of the singular fibre has singularities corresponding to the singularities in the fibre. In this note, we show that in the even relative-dimensional case, there exist elements of the bivariant Chow group of the base-change with supports in the singularities and hence their images in the bivariant Chow group with supports in the special fibre and that the usual cohomological vanishing cycles are obtained as their images by a natural map, a kind of “cycle map” so that the elements in the bivariant Chow groups can be regarded as the vanishing cycles. The bivariant Chow group with supports in the special fibre has a ring structure and the natural map is a ring homomorphism to the cohomology ring of the geometric generic fibre. Also discussed is the relation of the bivariant Chow group with supports in the special fibre to the specialization map of Chow groups.