Osculation for conic fibrations.

Smooth projective surfaces fibered in conics over a smooth curve are investigated with respect to their k-th osculatory behavior. Due to the bound for the dimension of their osculating spaces they do not differ at all from a general surface for k = 2, while their structure plays a significant role for k >= 3. The dimension of the osculating space at any point is studied taking into account the possible existence of curves of low degree transverse to the fibers, and several examples are discussed to illustrate concretely the various situations arising in this analysis. As an application, a complete description of the osculatory behavior of Castelnuovo surfaces is given. The case k = 3 for del Pezzo surfaces is also discussed, completing the analysis done for k = 2 in a previous paper by the authors (2001). Moreover, for conic fibrations X subset of P-N whose k-th inflectional locus has the expected codimension, a precise description of this locus is provided in terms of Chern classes. In particular, for N = 8, it turns out that either X is hypo-osculating for k = 3, or its third inflectional locus is 1-dimensional

E-polynomial of the SL(3,C)-character variety of free groups.

We compute the E-polynomial of the character variety of representations of a rank r free group in SL(3,C). Expanding upon techniques of Logares, Muñoz and Newstead (Rev. Mat. Complut. 26:2 (2013), 635-703), we stratify the space of representations and compute the E-polynomial of each geometrically described stratum using fibrations. Consequently, we also determine the E-polynomial of its smooth, singular, and abelian loci and the corresponding Euler characteristic in each case. Along the way, we give a new proof of results of Cavazos and Lawton (Int. J. Math. 25:6 (2014), 1450058).

E-polynomials of the SL(2, C)-character varieties of surface groups

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a once-punctured surface of any genus into SL(2, C), for any possible holonomy around the puncture. We follow the geometric technique introduced in [12], based on stratifying the space of representations, and on the analysis of the behavior of the E-polynomial under fibrations.

Deformations of canonical triple covers

In this paper, we show that if X is a smooth variety of general type of dimension m≥3 for which the canonical map induces a triple cover onto Y, where Y is a projective bundle over P1 or onto a projective space or onto a quadric hypersurface, embedded by a complete linear series (except Q3 embedded in P4), then the general deformation of the canonical morphism of X is again canonical and induces a triple cover. The extremal case when Y is embedded as a variety of minimal degree is of interest, due to its appearance in numerous situations. For instance, by looking at threefolds Y of minimal degree we find components of the moduli of threefolds X of general type with KX3=3pg−9,KX3≠6, whose general members correspond to canonical triple covers. Our results are especially interesting as well because they have no lower dimensional analogues.

Deformations of canonical double covers

In this paper we show that if X is a smooth variety of general type of dimension m≥2 for which its canonical map induces a double cover onto Y, where Y is the projective space, a smooth quadric hypersurface or a smooth projective bundle over P1, embedded by a complete linear series, then the general deformation of the canonical morphism of X is again canonical and induces a double cover. The second part of the article proves the non-existence of canonical double structures on the rational varieties above mentioned. Our results have consequences for the moduli of varieties of general type of arbitrary dimension, since they show that infinitely many moduli spaces of higher dimensional varieties of general type have an entire “hyperelliptic” component. This is in sharp contrast with the case of curves or surfaces of lower Kodaira dimension.

On the size of the fibers of spectral maps induced by semialgebraic embeddings

Let S(M) be the ring of (continuous) semialgebraic functions on a semialgebraic set M and S*(M) its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps Spec(j)1:Spec(S(N))→Spec(S(M)) and Spec(j)2:Spec(S*(N))→Spec(S*(M)) induced by the inclusion j:N M of a semialgebraic subset N of M. The ring S(M) can be understood as the localization of S*(M) at the multiplicative subset WM of those bounded semialgebraic functions on M with empty zero set. This provides a natural inclusion iM:Spec(S(M)) Spec(S*(M)) that reduces both problems above to an analysis of the fibers of the spectral map Spec(j)2:Spec(S*(N))→Spec(S*(M)). If we denote Z:=ClSpec(S*(M))(M N), it holds that the restriction map Spec(j)2|:Spec(S*(N)) Spec(j)2-1(Z)→Spec(S*(M)) Z is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of Spec(j)2 at the points of Z. The size of the fibers of prime ideals "close" to the complement Y:=M N provides valuable information concerning how N is immersed inside M. If N is dense in M, the map Spec(j)2 is surjective and the generic fiber of a prime ideal p∈Z contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber Spec(j)2-1(p) is a finite set for p∈Z. If such is the case, our procedure allows us to compute the size s of Spec(j)2-1(p). If in addition N is locally compact and M is pure dimensional, s coincides with the number of minimal prime ideals contained in p. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Mackey topology on locally convex spaces and on locally quasi-convex groups. Similarities and historical remarks

A counterpart of the Mackey–Arens Theorem for the class of locally quasi-convex topological Abelian groups (LQC-groups) was initiated in Chasco et al. (Stud Math 132(3):257–284, 1999). Several authors have been interested in the problems posed there and have done clarifying contributions, although the main question of that source remains open. Some differences between the Mackey Theory for locally convex spaces and for locally quasi-convex groups, stem from the following fact: The supremum of all compatible locally quasi-convex topologies for a topological abelian group G may not coincide with the topology of uniform convergence on the weak quasi-convex compact subsets of the dual groupG∧. Thus, a substantial part of the classical Mackey–Arens Theorem cannot be generalized to LQC-groups. Furthermore, the mentioned fact gives rise to a grading in the property of “being a Mackey group”, as defined and thoroughly studied in Díaz Nieto and Martín-Peinador (Proceedings in Mathematics and Statistics 80:119–144, 2014). At present it is not known—and this is the main open question—if the supremum of all the compatible locally quasi-convex topologies on a topological group is in fact a compatible topology. In the present paper we do a sort of historical review on the Mackey Theory, and we compare it in the two settings of locally convex spaces and of locally quasi-convex groups. We point out some general questions which are still open, under the name of Problems.

E-polynomial of SL(2,C)-character varieties of complex curves of genus 3.

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve of genus g = 3 into SL(2, C), and also of the moduli space of twisted representations. The case of genus g = 1, 2 has already been done in [12]. We follow the geometric technique introduced in [12], based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.

Spontaneous Meta-Arithmetic as the First Step Toward School Algebra. 'La meta-aritmética espontánea como el primer paso hacia el álgebra escolar'.

Resumen basado en el de la publicación

Compendio de la geometria elementar, aritmentica inferior, y trigonometria plana, y espherica /

Mode of access: Internet.

Arithmetica inferior y geometria practica y especulativa : origen de los nacimientos de las aguas dulces y gordas de esta Coronada Villa de Madrid... /

Mode of access: Internet.

Libro de algebra en arithmetica y geometria /

Mode of access: Internet.

Tractado subtilissimo d[e] Arismetica y de Geometria /

Mode of access: Internet.

Adaptações da geometria da coluna vertebral e do dorso durante a corrida

Universidade Estadual de Campinas . Faculdade de Educação Física

Efeito da geometria dos dentes da serra de fita na produção de madeira serrada de eucalipto

O objetivo deste trabalho foi determinar o efeito dos parâmetros que compõem os dentes de serra de fita na produção e qualidade da madeira serrada de eucalipto. Analisou-se o efeito de dois passos de dentes (57,15 mm e passo variado: 50,8 - 50,8 - 50,39 - 50,71 - 60,03 - 60,35 mm), de dois ângulos de ataque (26 e 270) e de duas alturas (22 e 25,4 mm) em duas classes diamétricas (25 a 34 cm e 35 e 47 cm) de clones de híbridos Eucalyptus grandis x Eucalyptus urophylla com 15 anos de idade. Não houve ganho de qualidade quando se trabalhou com passo variado. As lâminas de serra que possuíam ângulo de ataque de 270 e passo único tanto com altura do dente de 22 mm quanto com 25,4 mm geraram desvios acima da espessura-meta. A menor variação em torno da espessura-meta foi obtida quando se utilizaram lâminas de serra de ângulo de ataque de 270 e altura do dente de 25,4 mm. O maior desvio ocorreu quando o desdobro foi efetuado com passo variado e ângulo de ataque de 260. A utilização de passo variado, de modo geral, foi o que apresentou maiores desvios na espessura das tábuas em relação à espessura desejada.