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Geometry of certain Brill–Noether locus on a very general sextic surface and Ulrich bundles

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Abstract

Let \(X \subset {\mathbb {P}}^3\) be a very general sextic surface over complex numbers. In this paper, we study certain Brill–Noether problems for moduli of rank 2 stable bundles on X and its relation with rank 2 weakly Ulrich and Ulrich bundles. In particular, we show the non-emptiness of certain Brill–Noether loci and using the geometry of the moduli and the notion of the Petri map on higher dimensional varieties, we prove the existence of components of expected dimension. We also give sufficient conditions for the existence of rank 2 weakly Ulrich bundles \({\mathcal {E}}\) on X with \(c_1({\mathcal {E}}) =5H\) and \(c_2 \ge 91\) and partially address the question of whether these conditions really hold. We then study the possible implication of the existence of an weakly Ulrich bundle in terms of non-emptiness of Brill–Noether loci. Finally, using the existence of rank 2 Ulrich bundles on X we obtain some more non-empty Brill–Noether loci and investigate the possibility of existence of higher rank simple Ulrich bundles on X.

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Notes

  1. A smooth projective variety \(X \subseteq {\mathbb {P}}^n\) is called ACM (Arithmetically Cohen–Macaulay) if its homogeneous coordinate ring \(S_X\) is Cohen–Macaulay, or equivalently, if \(\text {dim}(S_X) = \text {depth}(S_X)\).

  2. A vector bundle \({\mathcal {E}}\) on a projective variety X is ACM if all its intermediate cohomologies vanishes, i.e., \(H^i(X, {\mathcal {E}}(m)) = 0\) for \(0<i <\mathrm{dim}(X)\) and all \(m \in {\mathbb {Z}}\).

  3. We say that \(X \subset {\mathbb {P}}^3\) of degree d can be defined by a linear pfaffian if there exists a \((2d)\times (2d)\) skew symmetric matrix M with linear entries such that \(X =\{\mathrm{pf}(M)=0\} \subset {\mathbb {P}}^3\).

  4. This can be seen using cohomological equivalences mentioned in Section 2 and the fact that if a line bundle has negative degree then it has no non-trivial global section.

  5. It is easy to see the following:

    • if \({\mathcal {E}}\) is a rank 2, \(\mu _H\)-stable vector bundle, then so is \({\mathcal {E}} \otimes {\mathcal {O}}_X(-2)\).

    • if \(c_1({\mathcal {E}}) = 5H, c_2({\mathcal {E}}) = c_2\), then using results from Section 2 we have \(c_1({\mathcal {E}} \otimes {\mathcal {O}}_X(-2)) = 5H-4H =H, c_2({\mathcal {E}} \otimes {\mathcal {O}}_X(-2)) = c_2 -36\).

    • the map in the opposite direction is given by \({\mathcal {F}} \mapsto {\mathcal {F}} \otimes {\mathcal {O}}_X(2)\), which makes f an isomorphism.

  6. This is because the trace free part is self dual, i.e., \(\mathrm{End}^0({\mathcal {G}})^* \cong \mathrm{End}^0({\mathcal {G}})\).

  7. Existence of such an \({\mathcal {E}}\) is guaranteed by Proposition 5.1.

  8. Indeed we have \(H^0({\mathcal {E}}^*_1 \otimes {\mathcal {E}}_2) \cong \mathrm{Hom}(\mathcal {E}_1 , {\mathcal {E}}_2)\) and it can be shown that both \({\mathcal {E}}_1, {\mathcal {E}}_2\) are of same slope 15. Therefore, there can be no non-trivial morphism between \({\mathcal {E}}_1\) and \({\mathcal {E}}_2\).

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Acknowledgements

The author would like to thank Dr. Sarbeswar Pal for many valuable comments. He thanks Dr. Krishanu Dan for answering a question on dimension estimate and Dr. Emre Coşkun for pointing out the relevant works regarding Ulrich bundles on surfaces. Finally, he would like to thank Prof. Luca Chiantini for pointing out the work on the classification of ACM bundles on general sextic surfaces.

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  1. Indian Institute of Science Education and Research,Thiruvananthapuram, Maruthamala PO, Vithura, Thiruvananthapuram, 695 551, India

    Debojyoti Bhattacharya

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Bhattacharya, D. Geometry of certain Brill–Noether locus on a very general sextic surface and Ulrich bundles. Proc Math Sci 132, 22 (2022). https://doi.org/10.1007/s12044-021-00652-5

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