Theta divisors and Ulrich bundles on geometrically ruled surfaces

  • Marian Aprodu
  • Gianfranco Casnati
  • Laura CostaEmail author
  • Rosa Maria Miró-Roig
  • Montserrat Teixidor I Bigas


We consider the following question: For which invariants g and e is there a geometrically ruled surface \(S \rightarrow C\) over a curve C of genus g with invariant e such that S is the support of an Ulrich line bundle with respect to a very ample line bundle? A surprising relation between the existence of certain proper theta divisors on some moduli spaces of vector bundles on C with the existence of Ulrich line bundles on S will be the key to completely solve the above question. The relation is realized by translating the vanishing conditions characterizing Ulrich line bundles to specific geometric conditions on the symmetric powers of the defining vector bundle of a given ruled surface. This general principle leads to some finer existence results of Ulrich line bundles in particular cases. Another focus is on the rank two case where, with very few exceptions, we show the existence of large families of special Ulrich bundles on arbitrary polarized ruled surfaces.


Vector bundle Ulrich bundle Geometrically ruled surface 

Mathematics Subject Classification

Primary 14J60 Secondary 14J26 



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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Facultatea de Matematică şi InformaticăUniversitatea din BucureştiBucureştiRomania
  2. 2.Institutul de Matematică “Simion Stoilow” al Academiei RomâneBucureştiRomania
  3. 3.Dipartimento di Scienze MatematichePolitecnico di TorinoTorinoItaly
  4. 4.Facultat de Matemàtiques i Informàtica, Departament de Matemàtiques i InformàticaBarcelonaSpain
  5. 5.Mathematics DepartmentTufts UniversityMedfordUSA

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