# writing

These are the things I have written over the past few years, all preprints are on the arXiv.

## Papers

### Published

- Embeddings of algebras in derived categories of surfaces, joint with Theo Raedschelders (arXiv:1501.04197),

published as*Proceedings of the American Mathematical Society, volume 145, number 7, 2017, pages 2757–2770* - The point variety of quantum polynomial rings, joint with Kevin De Laet and Lieven Le Bruyn (arXiv:1509.07312),

published as*Journal of Algebra, volume 463, 2016, pages 10–22* - Derived categories of noncommutative quadrics and Hilbert squares, joint with Theo Raedschelders (arXiv:1605.02795),

published online ahead of print in*International Mathematics Research Notices* - Relative tensor triangular Chow groups for coherent algebras, joint with Sebastian Klein (arXiv:1607.03423),

published as*Journal of Algebra, volume 487, 2017, pages 386–428* - Hochschild cohomology of noncommutative planes and quadrics (arXiv:1705.06098),

published as*Journal of Noncommutative Geometry, volume 13, 2019, pages 769–795* - Constructing noncommutative surfaces with exceptional collections of length 4, joint with Dennis Presotto (arXiv:1705.06943),

published as*Journal of the London Mathematical Society, volume 98, 2018, pages 85–103* - Examples violating Golyshev's canonical strip hypothesis, joint with Sergey Galkin and Swarnava Mukhopadhyay (arXiv:1806.07648, see also the related repositories canonical-strip-moduli-of-vector-bundles and ehrhart-polynomials-toric-fanos)

published online ahead of print in*Experimental Mathematics* - Admissible subcategories in derived categories of moduli of vector bundles on curves, joint with Swarnava Mukhopadhyay (arXiv:1807.00216)

published as*Advances in Mathematics, volume 351, 2019, pages 653–675* - Hilbert squares: derived categories and deformations, joint with Lie Fu and Theo Raedschelders (arXiv:1810.11873)

published as*Selecta Mathematica, volume 25, 2019, paper 37* - Automorphisms of Hilbert schemes of points on surfaces, joint with Georg Oberdieck and Jørgen Vold Rennemo (arXiv:1907.07064)

published online ahead of print in*Transactions of the American Mathematical Society*

### Preprints

- Comparison of two constructions of noncommutative surfaces with exceptional collections of length 4, joint with Dennis Presotto and Michel Van den Bergh (arXiv:1811.08810)
We will retitle this to "The Hirzebruch isomorphism for exotic noncommutative surfaces".

- Derived categories of (nested) Hilbert schemes, joint with Andreas Krug (arXiv:1909.04321)
- Hochschild cohomology of generalised Grassmannians, joint with Maxim Smirnov (arXiv:1911.09414)
- Moduli spaces of semiorthogonal decompositions in families, joint with Shinnosuke Okawa and Andrea Ricolfi (arXiv:2002.03303)
- Derived categories of standard flips and cubic hypersurfaces, joint with Lie Fu and Theo Raedschelders (arXiv:2002.04940)
- Indecomposability of derived categories in families, joint with Francesco Bastianelli, Shinnosuke Okawa and Andrea Ricolfi

### Work in progress

- Spanning classes on curves, joint with Swarnava Mukhopadhyay

### Other

- Semiorthogonal decompositions for moduli of sheaves on curves, Oberwolfach Reports 24/2018, Oberwolfach Workshop 1822, Interactions between Algebraic Geometry and Noncommutative Algebra, pages 1473–1476
- appendix "Comparing definitions of Hochschild cohomology for algebraic stacks" to The Hochschild cohomology ring of a global quotient orbifold, by Cris Negron and Travis Schedler (arXiv:1809.08715)

published online ahead of print in*Advances in Mathematics* - Artin–Schelter regular algebras versus smooth projective varieties (short note, to be extended at some point)

## Reviews

I have written reviews for## Notes

You might also be interested in my blog.### 2018

- Hochschild (co)homology, and the Hochschild–Kostant–Rosenberg decomposition, lecture notes for Advanced Topics in Algebra

### 2017

- Hochschild–Kostant–Rosenberg decomposition for surfaces, dimensions of components in the Hochschild–Kostant–Rosenberg decomposition for surfaces relevant to deformation theory

### 2016

- Segre symbols, a short note defining Segre symbols as a method of classifying base loci of pencils of quadric hypersurfaces
- The restriction of a homotopy-injective complex to a Zariski open subset is not necessarily homotopy-injective, details for the construction of such a complex, based on an explanation by Leonid Positselski
- Cheat sheet for mutations, because remembering where left and right goes is difficult

### 2015

- Cheat sheet for the simplicial indexing category, in case you always forget the notation (like me)
- Cheat sheet for recollements, the definition of a recollement and how to remember it
- The torsor for a formally smooth morphism, rough notes related to my lecture on lifting along a smooth morphism for ANAGRAMS about Hodge-to-de Rham degeneration

### 2014

- Spectral sequences: examples in algebra and algebraic geometry, notes for my lecture for ANAGRAMS about spectral sequences
- Quivers of exceptional collections on smooth projective varieties, poster for the British Algebraic Geometry meeting
- Positselski's main lemma, a short note elaborating on a lemma of Positselski that recognises vector bundles in the derived category
- Stratification of triangulated categories, notes for the lecture series by Henning Krause at the summer school on derived categories in Nantes (with corrections by Henning Krause, thanks!)
- Depth, and measuring singularities, notes for my short lecture for ANAGRAMS in the series on dimension functions
- My lecture series on Grothendieck duality for ANAGRAMS, also available as a single set of notes:
- Riemann–Roch theorem and Serre duality, notes for the first lecture
- More on Riemann–Roch and Serre duality, with applications, notes for the second lecture
- Derived categories and Grothendieck duality, notes for the third lecture
- Sketches of some of the proofs, notes for the fourth lecture

### 2013

- The $(\infty,1)$-topos of derived stacks, notes for my lecture at the Winter school on derived algebraic geometry
- Understanding $\mathbf{D}^{\mathrm{b}}(kQ)$ using moduli spaces, slides for my lecture at Young researchers in representation theory
- The triangulated structure on the category of projectives, notes for my lecture at the Summer school on preprojective algebras
- The derived Torelli theorem, notes for my lectures at the summer seminar in algebraic geometry of Columbia University and my exam for
*Surfaces K3 et théorie des déformations* - Grothendieck topologies and étale cohomology, notes for my lecture for the course
*Fonctions zêta et L de variétés et de motifs* - The derived stack of perfect complexes, notes for my lecture in Autour de la Géométrie Algébrique Dérivée, subsumed in the notes of the seminar
- Differential graded categories, notes for my lecture in Autour de la Géométrie Algébrique Dérivée, subsumed in the notes of the seminar

## Non-mathematical writing

- The fugue and the guitar (in Dutch)