In order to study the behavior of the points in a tower of curves, we introduce and study trivial points on towers of curves, and we discuss their finiteness over number fields. We relate the problem of proving that the only rational points are the trivial ones at some level of the tower, to the unboundeness of the gonality of the curves in the tower, which we show under some hypothesis.

We develop a new approach of extension calculus in the category of strict polynomial functors, based on Troesch complexes. We obtain new short elementary proofs of numerous classical $\mathrm{Ext}$-computations as well as new results. In particular, we get a cohomological version of the “fundamental theorems” from classical invariant theory for $G{L}_n$ for $n$ big enough (and we give a conjecture for smaller values of $n$). We also study the “twisting spectral sequence” $E^{s,t}(F,G,r)$ converging to the extension groups ${\mathrm{Ext}}_{{𝒫}_{𝕜}}^{*}(F^{\left(r\right)},G^{\left(r\right)})$ between the...

We define an intersection product of tropical cycles on tropical linear spaces $L_k^n$, i.e. on tropical fans of the type max${\{0,x_1,...,x_n\}}^{n-k}·{ℝ}^n$. Afterwards we use this result to obtain an intersection product of cycles on every smooth tropical variety, i.e. on every tropical variety that arises from gluing such tropical linear spaces. In contrast to classical algebraic geometry these products always yield well-defined cycles, not cycle classes only. Using these intersection products we are able to define the pull-back...

Classes dual to Schubert cycles constitute a basis on the cohomology ring of the flag manifold F, self-adjoint up to indexation with respect to the intersection form. Here, we study the bilinear form (X,Y) :=〈X·Y, c(F)〉 where X,Y are cocycles, c(F) is the total Chern class of F and〈,〉 is the intersection form. This form is related to a twisted action of the symmetric group of the cohomology ring, and to the degenerate affine Hecke algebra. We give a distinguished basis for this form, which is a...

Let $X$ be a normal projective variety, and let $A$ be an ample Cartier divisor on $X$. Suppose that $X$ is not the projective space. We prove that the twisted cotangent sheaf ${\Omega }_X\otimes A$ is generically nef with respect to the polarisation $A$. As an application we prove a Kobayashi-Ochiai theorem for foliations: if $ℱ⊊T_X$ is a foliation such that $detℱ\equiv i_{ℱ}A$, then $i_{ℱ}$ is at most the rank of $ℱ$.

For the Grothendieck group of a split simple linear algebraic group, the twisted γ-filtration provides a useful tool for constructing torsion elements in -rings of twisted flag varieties. In this paper, we construct a non-trivial torsion element in the γ-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline.

The theory of slice-regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains $\Omega$ of ${ℝ}^4$. When $\Omega$ is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which $\Omega$ is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space $ℂP^3$.

In this paper we investigate Hesse’s elliptic curves $H_{\mu }:U^3+V^3+W^3=3\mu UVW,\mu \in \mathbf{Q}-\left\{1\right\}$, and construct their twists, $H_{\mu ,t}$ over quadratic fields, and $\tilde{H}(\mu ,t),\mu ,t\in \mathbf{Q}$ over the Galois closures of cubic fields. We also show that $H_{\mu }$ is a twist of $\tilde{H}(\mu ,t)$ over the related cubic field when the quadratic field is contained in the Galois closure of the cubic field. We utilize a cubic polynomial, $R(t;X):=X^3+tX+t,t\in \mathbf{Q}-\{0,-27/4\}$, to parametrize all of quadratic fields and cubic ones. It should be noted that $\tilde{H}(\mu ,t)$ is a twist of $H_{\mu }$ as algebraic curves because it may not always have any rational points...