Severi varieties and Brill–Noether theory of curves on abelian surfaces

<jats:p>Severi varieties and Brill–Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface <jats:italic>S</jats:italic> with polarization <jats:italic>L</jats:italic> of type <jats:inline-formula id="j_crelle-2016-0029_ineq_9999"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_1262. png" /> <jats:tex-math>{(1,n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, we prove nonemptiness and regularity of the Severi variety parametrizing δ-nodal curves in the linear system <jats:inline-formula id="j_crelle-2016-0029_ineq_9998"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_2155. png" /> <jats:tex-math>{|L|}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for <jats:inline-formula id="j_crelle-2016-0029_ineq_9997"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_1301. png" /> <jats:tex-math>{0\leq\delta\leq n-1=p-2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (here <jats:italic>p</jats:italic> is the arithmetic genus of any curve in <jats:inline-formula id="j_crelle-2016-0029_ineq_9996"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_2155.png" /> <jats:tex-math>{|L|}</jats:tex-math> </jats:alternatives> </jats:inline-formula>). We also show that a general genus <jats:italic>g</jats:italic> curve having as nodal model a hyperplane section of some <jats:inline-formula id="j_crelle-2016-0029_ineq_9995"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_1262. png" /> <jats:tex-math>{(1,n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many <jats:inline-formula id="j_crelle-2016-0029_ineq_9994"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_1262. png" /> <jats:tex-math>{(1,n)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-polarized abelian surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is proved concerning the possibility of deforming a genus <jats:italic>g</jats:italic> curve in <jats:italic>S</jats:italic> equigenerically to a nodal curve. The rest of the paper deals with the Brill–Noether theory of curves in <jats:inline-formula id="j_crelle-2016-0029_ineq_9993"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_2155. png" /> <jats:tex-math>{|L|}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. It turns out that a general curve in <jats:inline-formula id="j_crelle-2016-0029_ineq_9992"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_2155. png" /> <jats:tex-math>{|L|}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is Brill–Noether general. However, as soon as the Brill–Noether number is negative and some other inequalities are satisfied, the locus <jats:inline-formula id="j_crelle-2016-0029_ineq_9991"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_2153. png" /> <jats:tex-math>{|L|^{r}_{d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of smooth curves in <jats:inline-formula id="j_crelle-2016-0029_ineq_9990"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_2155. png" /> <jats:tex-math>{|L|}</jats:tex-math> </jats:alternatives> </jats:inline-formula> possessing a <jats:inline-formula id="j_crelle-2016-0029_ineq_9989"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_1899. png" /> <jats:tex-math>{g^{r}_{d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is nonempty and has a component of the expected dimension. As an application, we obtain the existence of a component of the Brill–Noether locus <jats:inline-formula id="j_crelle-2016-0029_ineq_9988"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_2109. png" /> <jats:tex-math>{{\mathcal{M}}^{r}_{p,d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> having the expected codimension in the moduli space of curves <jats:inline-formula id="j_crelle-2016-0029_ineq_9987"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_2111. png" /> <jats:tex-math>{{\mathcal{M}}_{p}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. For <jats:inline-formula id="j_crelle-2016-0029_ineq_9986"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_crelle-2016-0029_eq_2000. png" /> <jats:tex-math>{r=1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, the results are generalized to nodal curves. </jats:p>