STABLE ULRICH BUNDLES

By Marta Casanellas, Robin Hartshorne. 2011

112 citations recorded by the source corpus.

The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of generators) on nonsingular cubic surfaces X ⊂ ℙ3. We give necessary and sufficient conditions on the first Chern class D for the existence of stable Ulrich bundles on X of rank r and c1 = D. When such bundles exist, we prove that the corresponding moduli space of stable bundles is smooth and irreducible of dimension D2 - 2r2 + 1 and consists entirely of stable Ulrich bundles (see Theorem 1.1). We are also able to prove the existence of stable Ulrich bundles of any rank on nonsingular cubic threefolds in ℙ4, and we show that the restriction map from bundles on the threefold to bundles on the surface is generically etale and dominant.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f992ee4f053351a6a16632c71e7624c361cc4213

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In Algebra and Number Theory

1. Rosa M. Mir'o-Roig Joan Pons-Llopis
2. Emre Coskun
3. Rajesh S. Kulkarni
4. G. Casnati
5. Yusuf Mustopa

In Geometry and Topology

1. G. Casnati
2. Rosa M. Mir'o-Roig Joan Pons-Llopis
3. Emre Coskun
4. Rajesh S. Kulkarni
5. Yusuf Mustopa

In Mathematical Physics

1. G. Casnati
2. Rosa M. Mir'o-Roig Joan Pons-Llopis
3. Emre Coskun
4. Rajesh S. Kulkarni
5. Yusuf Mustopa

Citations in the corpus

• Special Ulrich bundles on non-special surfaces with $p_g=q=0$ (2016, background): “omposable, Ulrich bundles for arbitrary large p. For example Ulrich–wildness (with respect to a suitable line bundle) is known for each anticanonically embedded surface of degree at least 3 (see e.g. [11], [15], [28], [32], [12]), all the Segre products but P1 ×P1 (see [18]), determinantal varieties”
• Pfaffian quartic surfaces and representations of Clifford algebras (2011, background): “cludes the proof of (ii). Remark 3.23. The theorems of Qin that we have just used refer to slope-stability, rather than stability. However, slope-stability implies stability, and by Theorem 2.9(c) of [CH] the two notions coincide for Ulrich bundles on any smooth projective variety. 4. ConstructingIr”
• THE GEOMETRY OF ULRICH BUNDLES ON DEL PEZZO SURFACES (2011, methodology): “adric surface case. The remainder of this section is devoted to the construction of Ulrich bundles on Xd of rank rH for 3 ≤d ≤7 and r ≥2. While this has already been done in the papers [CH],[MP2] and [CH1], we present an alternate construction based on Theorem 1.1. Proposition 3.7. Suppose 3 ≤d ≤7. ”
• n-dimensional Fano varieties of wild representation type (2010, methodology): “In [5], it is also proved that smooth cubic surfaces are of wild representation type.”