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.

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.