102,436 research outputs found
On the arithmetic of del Pezzo surfaces of degree 2
No full textWe study the arithmetic of certain del Pezzo surfaces of degree 2. We produce examples of Brauer-Manin obstruction to the Hasse principle, coming from 2- and 4-torsion elements in the Brauer group
Birational Geometry of 3-fold Mori Fibre Spaces
No full textWe study the geography and birational geometry of 3-fold conic bundles over P and cubic del Pezzo fibrations over P. We discuss many explicit examples and raise several open questions
Groups acting freely on Calabi-Yau threefolds embedded in products of del Pezzo surfaces
No full textIn this paper, we investigate quotients of Calabi-Yau manifolds Y embedded in Fano varieties X, which are products of two del Pezzo surfaces - with respect to groups G that act freely on Y. In particular, we revisit some known examples and we obtain some new Calabi-Yau varieties with small Hodge numbers. The groups G are subgroups of the automorphism groups of X, which is described in terms of the automorphism group of the two del Pezzo surfaces
Bounding geometrically integral del Pezzo surfaces
No full textWe prove several boundedness statements for geometrically integral normal del Pezzo surfaces X over arbitrary fields. We give an explicit sharp bound on the irregularity if X is canonical or regular. In particular, we show that wild canonical del Pezzo surfaces exist only in characteristic
. As an application, we deduce that canonical del Pezzo surfaces form a bounded family over
, generalising work of Tanaka. More generally, we prove the BAB conjecture on the boundedness of
-klt del Pezzo surfaces over arbitrary fields of characteristic different from
and
G-torsors and universal torsors over nonsplit del Pezzo surfaces
No full textLet S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of in Pic , or a form of it containing the Néron-Severi torus. Let be the G-torsor over obtained by extension of structure group from a universal torsor over . We prove that does not descend to S unless does. This is in contrast to a result of Friedman and Morgan that such always descend to singular del Pezzo surfaces over from their desingularizations.12 page
On -birational rigidity of del Pezzo surfaces
No full textLet be a finite group and be its subgroup. We prove that if a smooth del Pezzo surface over an algebraically closed field is -birationally rigid then it is also -birationally rigid, answering a geometric version of Koll\'{a}r's question in dimension 2 by positive
On -birational rigidity of del Pezzo surfaces
No full textLet be a finite group and be its subgroup. We prove that
if a smooth del Pezzo surface over an algebraically closed field is
-birationally rigid then it is also -birationally rigid, answering a
geometric version of Koll\'{a}r's question in dimension 2 by positive.Comment: 27 pages. Substantially updated version, main results unchanged. A
new section is added, plus many explicit example
Symmetries of Del Pezzo Surfaces
No full textFor each del Pezzo surface X of degree d, the action of the automorphism group Aut(X) on the exceptional curves of X induces a map ρ : Aut(X) → W(Rd), where W(Rd) is the Weyl group of a root system Rd dependent on the degree of X. The image of ρ is well defined up to conjugacy in W(Rd), so we say that a group G acts by automorphisms on a del Pezzo surface X of degree d if a representative of the conjugacy class of G in W(Rd) is contained in ρ(Aut(X)).
The purpose of this thesis is to determine, for each field k of characteristic zero, and for each degree d ≥ 3, the subgroups of W(Rd) that act by automorphisms on a del Pezzo surface of degree d over k. We also provide explicit equations for del Pezzo surfaces of degrees 3 and 4 that admit these group actions, and we determine which of these surfaces are k-rational, stably k-rational, or k-unirational. If a group G acts on a k-rational del Pezzo surface, there is an associated embedding ι : G → Cr2(k) into the plane Cremona group over k. In this way, we make progress toward a classification of the finite subgroups of Cr2(k) over any field of characteristic zero
The maximal number of singular points on log del Pezzo surfaces
No full textAbstract. We prove that a del Pezzo surface with Picard number one has at most four singular points. Intoduction A log del Pezzo surface is a projective algebraic surface X with only quotient singularities and ample anticanonical divisor −K X . Del Pezzo surfaces naturally appear in the log minimal model program (see, e. g., Recall that a normal complex projective surface is called a rational homology projective plane if it has the same Betti numbers as the projective plane P 2 . J. Kollár [9] posed the problem to classify rational homology P 2 's with quotient singularities having five singular points. In [4] this problem is solved for the case of numerically effective K X . Our main theorem solves Kollár's problem in the case where −K X is ample. The author is grateful to Professor Y. G. Prokhorov for suggesting this problem and for his help. The author also would like to thank the referee for useful comments
Automorphisms of quartic del Pezzo surfaces in characteristic zero
No full textFor each field of characteristic zero, we classify which groups act by
automorphisms on a quartic del Pezzo surface over . We also determine which
groups act on -rational, stably -rational, or -unirational quartic del
Pezzo surfaces. For each group that is realized over , we exhibit
explicit equations for a quartic del Pezzo surface in such
that acts by automorphisms on .Comment: 22 page
- …
