There is a close relationship between the embedded topology of complex plane curves and (group-theoretic) arithmetic elliptic curves. In recent paper, we studied some arrangements that include special smooth component, via torsion properties induced by divisors in curve associated to remaining components, which an property. When this has maximal flexes, there natural isomorphism its Jacobian va...

We use functoriality of tropicalization and the geometry of projections of subvarieties of tori to show that the fibers of the tropicalization map are dense in the Zariski topology. For subvarieties of tori over fields of generalized power series, points in each tropical fiber are obtained “constructively” using Kedlaya’s transfinite version of Newton’s method.

The polar curves of foliations F having a curve C of separatrices generalize the classical polar curves associated to hamiltonian foliations of C. As in the classical theory, the equisingularity type ℘(F) of a generic polar curve depends on the analytical type of F , and hence of C. In this paper we find the equisingularity types ǫ(C) of C, that we call kind singularities, such that ℘(F) is com...

1 Affine schemes 4 1.1 Motivation and review of varieties . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 First attempt at defining an affine scheme . . . . . . . . . . . . . . . . . . . 4 1.3 Affine schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 The Zariski topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 The ringed space s...

A meadow is a zero totalised field (0 = 0), and a cancellation meadow is a meadow without proper zero divisors. In this paper we consider differential meadows, i.e., meadows equipped with differentiation operators. We give an equational axiomatization of these operators and thus obtain a finite basis for differential cancellation meadows. Using the Zariski topology we prove the existence of a d...

×ØÖÖغ In the present paper, we present a slightly strengthened version of a well-known theorem of Belyi on the existence of " Belyi maps ". Roughly speaking, this strengthened version asserts that there exist Belyi maps which are unramified at [cf. Theorem 2.5] — or even near [cf. Corollary 3.2] — a prescribed finite set of points. Write C for the complex number field; Q ⊆ C for the subfield o...

We show that if a group automorphism of Cremona arbitrary rank is also homeomorphism with respect to either the Zariski or Euclidean topology, then it inner up field base-field. Moreover, we similar result holds consider groups polynomial automorphisms affine spaces instead groups.