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If $V_{d,\delta }$ denotes the variety of irreducible plane curves of degree $d$ with exactly $\delta$ nodes as singularities, Diaz and Harris (1986) have conjectured that $\mathrm{Pic}\left(V_{d,\delta }\right)$ is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that $\mathrm{Pic}\left(V_{d,1}\right)$ is a finite group, so that the conjecture holds for $\delta =1$. Actually the order of $\mathrm{Pic}\left(V_{d,1}\right)$ is $6(d-2)d^2-3d+1)$, the group being cyclic if $d$ is odd and the product of ${\mathbb{Z}}_2$ and a cyclic group of order $3(d-2)(d^2-3d+1)$ if $d$ is even.

Given an elliptic curve E and a finite subgroup G, Vélu's formulae concern to a separable isogeny I_G: E → E' with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P+G as the difference between the abscissa of I_G(P) and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstrass...

This paper is devoted to the study of the volcanoes of ℓ-isogenies of elliptic curves over a finite field, focusing on their height as well as on the location of curves across its different levels. The core of the paper lies on the relationship between the ℓ-Sylow subgroup of an elliptic curve and the level of the volcano where it is placed. The particular case ℓ = 3 is studied in detail, giving an algorithm to determine the volcano of 3-isogenies of a given elliptic curve. Experimental results...