1,721,055 research outputs found
Projective techniques in twistor geometry
Full text linkThis survey explores the interplay between twistor geometry and projective geometry, focusing on their applications to algebraic surfaces. We explore two main topics: the inclusion of twistor fibers and lines in these surfaces, and the behavior of twistor discriminant loci, with a particular focus on degree-2 surfaces. The study highlights contributions from Ballico and collaborators, comparing the twistor spaces of CP3 and the flag threefold F, which reveal fascinating contrasts and parallels. Key findings include a detailed analysis of surfaces in CP3 and F that either contain finite or infinite twistor fibers. The survey also touches on cubic surfaces in CP3 and their counterparts in F, where the configurations of twistor fibers lead to intriguing results. Special attention is given to surfaces of twistor degree 2, including how their geometry and singularities interact with twistor projections. In particular, we discuss smooth and singular surfaces in F of bidegree (1, 1) and (0, 2), as well as their discriminant loci
On the real differential of a slice regular function
No full textAbstract
In this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theorem proved by G. Gentili, S. Salamon and C. Stoppato and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice regular functions (called spherical expansion), and to a new general result which says that the slice derivative of any injective slice regular function is different from zero. A useful tool proven in this paper is a new formula that relates slice and spherical derivatives of a slice regular function. Given a slice regular function, part of its singular set is described as the union of surfaces on which it results to be constant.</jats:p
Twistor interpretation of slice regular functions
No full textGiven a slice regular function f:Ω⊂H→H, with Ω∩R≠∅, it is possible to lift it to surfaces in the twistor space CP3 of S4≃H∪{∞} (see Gentili et al., 2014). In this paper we show that the same result is true if one removes the hypothesis Ω∩R≠∅ on the domain of the function f. Moreover we find that if a surface S⊂CP3 contains the image of the twistor lift of a slice regular function, then S has to be ruled by lines. Starting from these results we find all the projective classes of algebraic surfaces up to degree 3 in CP3 that contain the lift of a slice regular function. In addition we extend and further explore the so-called twistor transform, that is a curve in Gr2(C4) which, given a slice regular function, returns the arrangement of lines whose lift carries on. With the explicit expression of the twistor lift and of the twistor transform of a slice regular function we exhibit the set of slice regular functions whose twistor transform describes a rational line inside Gr2(C4), showing the role of slice regular functions not defined on R. At the end we study the twistor lift of a particular slice regular function not defined over the reals. This example shows the effectiveness of our approach and opens some questions
Three Topological Results on the Twistor Discriminant Locus in the 4-Sphere
No full textWe exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration π: CP 3 → S 4 . We prove three results about the topology of the twistor discriminant locus of an algebraic surface in CP 3 . First of all we prove that, with the exception of two special cases, the real dimension of the twistor discriminant locus of an algebraic surface is always equal to 2. Secondly we describe the possible intersections of a general surface with the family of twistor lines: we find that only 4 configurations are possible and for each of them we compute the dimension. Lastly we give a decomposition of the twistor discriminant locus of a given cone in terms of its singular locus and its dual variety
Slice-polynomial functions and twistor geometry of ruled surfaces in CP 3
Full text linkIn the present paper we introduce the class of slice-polynomial functions: slice regular functions defined over the quaternions, outside the real axis, whose restriction to any complex half-plane is a polynomial. These functions naturally emerge in the twistor interpretation of slice regularity introduced in Gentili et al. (J Eur Math Soc 16(11):2323–2353, 2014) and developed in Altavilla (J Geom Phys 123:184–208, 2018). To any slice-polynomial function P we associate its companionP ∨ and its extension to the real axis P R , that are quaternionic functions naturally related to P. Then, using the theory of twistor spaces, we are able to show that for any quaternion q the cardinality of simultaneous pre-images of q via P, P ∨ and P R is generically constant, giving a notion of degree. With the brand new tool of slice-polynomial functions, we compute the twistor discriminant locus of a cubic scroll C in CP 3 and we conclude by giving some qualitative results on the complex structures induced by C via the twistor projection
La tessera europea d'assicurazione malattia: verso una maggiore integrazione nel settore sanitario
No full textTwistor lines on algebraic surfaces
No full textWe give quantitative and qualitative results on the family of surfaces in CP 3 containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines E. We prove that its general element is a smooth surface containing E and no other line. Afterward we prove that twistor lines are Zariski dense in the Grassmannian Gr(2, 4). Then, for any degree d≥ 4 , we give lower bounds on the maximum number of twistor lines contained in a degree d surface. The smooth and singular cases are studied as well as the j-invariant one
Systemic Governance and accountability. Working and re-working the conceptual and spatial boudnaries
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