# 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 as*IMRN, volume 2020, issue 19, pages 6042–6069* - 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 as*Transactions of the American Mathematical Society, volume 373 (9), 2020, pages 6139–6156* - Derived categories of the Cayley plane and the coadjoint Grassmannian in type F, joint with Alexander Kuznetsov and Maxim Smirnov (arXiv:2005.01989)

published online ahead of print in Transformation Groups

### 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 (arXiv:2007.00994)
- Graph potentials and topological quantum field theories, joint with Sergey Galkin and Swarnava Mukhopadhyay (arXiv:2009.05568)
- Polyvector fields for Fano 3-folds, joint with Enrico Fatighenti and Fabio Tanturri (arxiv:2104.07626)
- Projectivity of the moduli space of vector bundles on a curve, joint with Jarod Alper, Daniel Bragg, Jason Liang and Tuomas Tajakka (expository article, preprint will be available soon)
- Graph potentials and symplectic geometry of moduli of vector bundles, joint with Sergey Galkin and Swarnava Mukhopadhyay (to be split off from arXiv:2009.05568, preprint available soon)
- Motivic, orthogonal and semiorthogonal decompositions of moduli of vector bundles, joint with Sergey Galkin and Swarnava Mukhopadhyay (to be split off from arXiv:2009.05568)
- A combinatorial non-abelian Torelli theorem, joint with Sergey Galkin and Swarnava Mukhopadhyay, preprint available soon

### Other publications

- 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 as*Advances in Mathematics, volume 364, 2020* - How do semiorthogonal decompositions behave in families?, preprint for Oberwolfach Reports, Oberwolfach Workshop 2029, Algebraic Geometry: Moduli Spaces, Birational Geometry and Derived Aspects, preprint

### Lecture notes

- 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!)
- My lecture series on Grothendieck duality for ANAGRAMS, as a single set of notes, or separately:
- 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

- Hochschild (co)homology, and the Hochschild–Kostant–Rosenberg decomposition, lecture notes for Advanced Topics in Algebra, a course from 2018 at the University of Bonn (to be expanded in the future)

### For a broader audience

- Het Stacks project, Nieuw Archief voor Wiskunde (5) 21 (2020), no. 3, 167–173 (pdf)
- Curating Online Mathematical Resources, to appear in the October issue of Notices of the American Mathematical Society

### Not intended for publication

- Segre symbols (short note defining Segre symbols as a method of classifying base loci of pencils of quadric hypersurfaces, which seems to have been of interest to some)
- Artin–Schelter regular algebras versus smooth projective varieties (short note, possibly to be extended at some point)

## Non-mathematical writing

- The fugue and the guitar (in Dutch)

## Reviews

I have written reviews for## Some other notes

You might also be interested in my blog, where various things that could also have been pdf's are written in html. The notes that are listed in what follows are less polished or less important (which is a vague opinion anyway) than the ones listed above as lecture notes or not intended for publication, or they are purely of expository and standard nature.

### 2021

- On $\int\mathrm{td}_X^{1/2}$ of O'Grady's exceptional hyperkähler varieties (short note on a computation, see also the accompanying code)

### 2017

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

### 2016

- 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
- Depth, and measuring singularities, notes for my short lecture for ANAGRAMS in the series on dimension functions

### 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