We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over  $\mathbb{Q}$ of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus 3 examples defined by polynomials with small coefficients.

Using the $\sharp$-minimal model program of uniruled varieties we show that, for any pair $(X, {\mathcal H})$ consisting of a reduced and irreducible variety $X$ of dimension $k \geq 3$ and a globally generated big line bundle ${\mathcal H}$ on $X$ with $d:= {\mathcal H}^k$ and $n:= h^0(X, {\mathcal H})-1$ such that $d<2(n-k)-4$, then $X$ is uniruled of ${\mathcal H}$-degree one, except if $(k,d,n)=(3,27,19)$ and a ${\sharp}$-minimal model of $(X, {\mathcal H})$ is $({\mathbb P}^3,{\mathcal O}_{{\mathbb P}^3}(3))$. We also show that the bound is optimal for threefolds.

We prove the unirationality of the moduli space of complex curves of genus 14. The method essentially relies on linkage of curves. In particular it is shown that a general curve of genus 14 admits a projective model D, in a six-dimensional projective space, which is linked to a general curve C of degree 14 and genus 8 by a complete intersection of quadrics. Using this property we are able to obtain, after a reasonable amount of further work, the unirationality result in the case of genus 14. Moreover, some variations of the same method, involving the Hilbert schemes of curves of very low genus, are used to obtain the same result for the known cases of genus 11, 12, 13.

three families of pairs of curves are presented; each pair consists of geometrically non-isomorphic curves whose jacobians are isomorphic to one another as unpolarized abelian varieties. each family is parametrized by an open subset of ${\mathbb p}^1$. the first family consists of pairs of genus-2 curves whose equations are given by simple expressions in the parameter; the curves in this family have reducible jacobians. the second family also consists of pairs of genus-2 curves, but generically the curves in this family have absolutely simple jacobians. the third family consists of pairs of genus-3 curves, one member of each pair being a hyperelliptic curve and the other a plane quartic. examples from these families show that in general it is impossible to tell from the jacobian of a genus-2 curve over $\mathbb q$ whether or not the curve has rational points – or indeed whether or not it has real points. the families are constructed using methods that depend on earlier joint work with franck leprévost and bjorn poonen, and on peter bending's explicit description of the curves of genus 2 whose jacobians have real multiplication by $\mathbb z[\sqrt{2}]$.