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Abelian varieties in \(W_ d^ r (C)\) and points of bounded degree on algebraic curves. (English) Zbl 0808.14025

Let \(C\) be a smooth complex curve of genus \(g\). The article answers some questions raised by Abramovich and Harris about abelian varieties \(A\) contained in the subvarieties \(W^ r_ d (C)\) of \(\text{Pic}^ d (C)\), defined as the set of line bundles of degree \(d\) with at least \(r+1\) independent sections. If \(d \leq g - 1 + r\), the dimension of \(A\) is at most \(d/2 - r\); if there is equality and \(d \leq 2 (g - 1 + r)/3\), then \(C\) is a double cover of a smooth curve \(B\) of genus \(d/2-r\), and \(A\) is the pull-back of \(\text{Pic}^{d/2} (B)\). The Prym construction shows that the latter statement may not hold if \(r=0\) and \(d=g-1\).
We also construct various counter-examples to the following statement by D. Abramovich and J. Harris [see Compos. Math. 78, No. 2, 227-238 (1991; Zbl 0748.14010)]: Suppose \(C\) is covered by a curve with a map of degree \(d\) or less onto a curve of genus \(h\) or less; then \(C\) has the same property. Finally, for each \(d \geq 4\), we construct a curve \(C\) defined over a number field \(K\) that has infinitely many points of degree at most \(d\) over \(K\), but no maps of degree \(d\) or less onto a rational or elliptic curve. There are no such curves for \(d=2\) or 3.

MSC:

14H60 Vector bundles on curves and their moduli
14K05 Algebraic theory of abelian varieties
14H30 Coverings of curves, fundamental group
14G05 Rational points

Citations:

Zbl 0748.14010
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References:

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