# On nodal Enriques surfaces and quartic double solids

@article{Ingalls2010OnNE, title={On nodal Enriques surfaces and quartic double solids}, author={Colin Ingalls and Alexander Kuznetsov}, journal={Mathematische Annalen}, year={2010}, volume={361}, pages={107-133} }

We consider the class of singular double coverings $$X \rightarrow {\mathbb {P}}^3$$X→P3 ramified in the degeneration locus $$D$$D of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such a quartic surface $$D,$$D, one can associate an Enriques surface $$S$$S which is the factor of the blowup of $$D$$D by a natural involution acting without fixed points (such Enriques…

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## References

SHOWING 1-10 OF 13 REFERENCES

### Scheme of lines on a family of 2-dimensional quadrics: geometry and derived category

- Mathematics
- 2010

Given a generic family $$Q$$Q of 2-dimensional quadrics over a smooth 3-dimensional base $$Y$$Y we consider the relative Fano scheme $$M$$M of lines of it. The scheme $$M$$M has a structure of a…

### Some Elementary Examples of Unirational Varieties Which are Not Rational

- Mathematics
- 1972

An outstanding problem in the algebraic geometry of varieties of dimension n ^ 3 over an algebraically closed field k has been whether there exist unirational varieties which are not rational. Here V…

### On analytic surfaces with double points

- MathematicsProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
- 1958

It is shown that the non-singular model of an algebraic surface, lying in complex projective 3-space and possessing only ordinary double points, is differentiably homeomorphic to any non-singular…

### Complex Algebraic Surfaces

- Mathematics, Chemistry
- 1996

Introduction Notation Part I. The Picard Group and the Riemann-Roch Theorem: Part II. Birational Maps: Part III. Ruled Surfaces: Part IV. Rational Surfaces: Part V. Castelnuovo's Theorem and…

### Reconstruction of a Variety from the Derived Category and Groups of Autoequivalences

- MathematicsCompositio Mathematica
- 2001

We consider smooth algebraic varieties with ample either canonical or anticanonical sheaf. We prove that such a variety is uniquely determined by its derived category of coherent sheaves. We also…

### REPRESENTABLE FUNCTORS, SERRE FUNCTORS, AND MUTATIONS

- Mathematics
- 1990

This paper studies the categorical version of the concept of mutations of an exceptional set, as used in the theory of vector bundles. The basic object of study is a triangulated category with a…

### PROJECTIVE BUNDLES, MONOIDAL TRANSFORMATIONS, AND DERIVED CATEGORIES OF COHERENT SHEAVES

- Mathematics
- 1993

This paper studies derived categories of coherent sheaves on varieties that are obtained by projectivization of vector bundles and by monoidal transformations. Conditions for the existence of…

### Representation Theory: A First Course

- Mathematics
- 1991

This volume represents a series of lectures which aims to introduce the beginner to the finite dimensional representations of Lie groups and Lie algebras. Following an introduction to representation…