John Christian Ottem

• Ottem, John Christian; Rennemo, Jørgen Vold & Kollár, János (2024). Fano varieties with torsion in the third cohomology group. Journal für die Reine und Angewandte Mathematik. ISSN 0075-4102. doi: 10.1515/crelle-2024-0020. Full text in Research Archive Show summary

  We construct first examples of Fano varieties with torsion in their third cohomology group. The examples are constructed as double covers of linear sections of rank loci of symmetric matrices, and can be seen as higher-dimensional analogues of the Artin–Mumford threefold. As an application, we answer a question of Voisin on the coniveau and strong coniveau filtrations of rationally connected varieties.

• Huybrechts, Daniel & Ottem, John Christian (2024). Nodal quintic surfaces and lines on cubic fourfolds. L'Enseignement mathématique (LEM). ISSN 0013-8584. 70(3/4), p. 499–532. doi: 10.4171/LEM/1063. Full text in Research Archive Show summary

  We study nodal quintic surfaces with an even set of 16 nodes as analogues of singular Kummer surfaces. The interpretation of the natural double cover of an even 16-nodal quintic as a certain Fano variety of lines could be viewed as a replacement for the additive structure of the cover of a singular Kummer surface by its associated abelian surface. Most of the results in this article can be seen as refinements of known facts and our arguments rely heavily on techniques developed by Beauville (1979), Murre (1972), and Voisin (1986), Results due to Shen (2012, 2014) are particularly close to some of the statements. In this sense, the text is mostly expository (but with complete proofs), although our arguments often differ substantially from the original sources.

• Degtyarev, Alex; Itenberg, Ilia & Ottem, John Christian (2023). Planes in cubic fourfolds. Algebraic Geometry. ISSN 2313-1691. 10(2), p. 228–258. doi: 10.14231/AG-2023-007. Full text in Research Archive Show summary

  We show that the maximal number of planes in a complex smooth cubic fourfold in P5 is 405, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is 357, realized by the so-called Clebsch–Segre cubic. Altogether, there are but three (up to projective equivalence) cubics with more than 350 planes.

• Ottem, John Christian & Suzuki, Fumiaki (2023). An O -acyclic variety of even index: With an appendix by Olivier Wittenberg. Mathematische Annalen. ISSN 0025-5831. doi: 10.1007/s00208-023-02581-2. Full text in Research Archive Show summary

  We give the first examples of O-acyclic smooth projective geometrically connected varieties over the function field of a complex curve, whose index is not equal to one. More precisely, we construct a family of Enriques surfaces over P1 such that any multi-section has even degree over the base P1 and show moreover that we can find such a family defined overQ. This answers affirmatively a question of Colliot-Thélène and Voisin. Furthermore, our construction provides counterexamples to: the failure of the Hasse principle accounted for by the reciprocity obstruction; the integral Hodge conjecture; and universality of Abel–Jacobi maps.

• Nicaise, Johannes & Ottem, John Christian (2022). Tropical degenerations and stable rationality. Duke mathematical journal. ISSN 0012-7094. 171(15), p. 3023–3075. doi: 10.1215/00127094-2022-0065. Full text in Research Archive Show summary

  We use the motivic obstruction to stable rationality introduced by Shinder and the first-named author to establish several new classes of stably irrational hypersurfaces and complete intersections. In particular, we show that very general quartic fivefolds and complete intersections of a quadric and a cubic in P6 are stably irrational. An important new ingredient is the use of tropical degeneration techniques.

• Nicaise, Johannes & Ottem, John Christian (2021). A refinement of the motivic volume, and specialization of birational types. Progress in Mathematics. ISSN 0743-1643. 342, p. 299–322. doi: 10.1007/978-3-030-75421-1. Full text in Research Archive Show summary

  We construct an upgrade of the motivic volume by keeping track of dimensions in the Grothendieck ring of varieties. This produces a uniform refinement of the motivic volume and its birational version introduced by Kontsevich and Tschinkel to prove the specialization of birational types. We also provide several explicit examples of obstructions to stable rationality arising from this technique.

• Benoist, Olivier & Ottem, John Christian (2021). Two coniveau filtrations. Duke mathematical journal. ISSN 0012-7094. 170(12), p. 2719–2753. doi: 10.1215/00127094-2021-0055. Full text in Research Archive Show summary

  A cohomology class of a smooth complex variety of dimension n has coniveau at least c if it vanishes in the complement of a closed subvariety of codimension at least c, and it has strong coniveau at least c if it comes by proper pushforward from the cohomology of a smooth variety of dimension at most n−c. We show that these two notions differ in general, both for integral classes on smooth projective varieties and for rational classes on smooth open varieties.

• Benoist, Olivier & Ottem, John Christian (2020). Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero. Commentarii Mathematici Helvetici. ISSN 0010-2571. 95, p. 27–35. doi: 10.4171/CMH/479. Full text in Research Archive Show summary

  We prove that the product of an Enriques surface and a very general curve of genus at least 1 does not satisfy the integral Hodge conjecture for 1-cycles. This provides the first examples of smooth projective complex threefolds of Kodaira dimension zero for which the integral Hodge conjecture fails, and the first examples of non-algebraic torsion cohomology classes of degree 4 on smooth projective complex threefolds.

• Ottem, John Christian & Suzuki, Fumiaki (2020). A pencil of Enriques surfaces with non-algebraic integral Hodge classes. Mathematische Annalen. ISSN 0025-5831. 377, p. 183–197. doi: 10.1007/s00208-020-01969-8. Full text in Research Archive Show summary

  We prove that there exists a pencil of Enriques surfaces defined over Q with non-algebraic integral Hodge classes of non-torsion type. This gives the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. As an application, we construct a fourfold which gives the negative answer to a classical question of Murre on the universality of the Abel-Jacobi maps in codimension three.

• Gounelas, Frank & Ottem, John Christian (2020). Remarks on the positivity of the cotangent bundle of a K3 surface. Épijournal de Géométrie Algébrique (EPIGA). 4(8). doi: 10.46298/epiga.2020.volume4.5924. Full text in Research Archive Show summary

  Using recent results of Bayer–Macrì, we compute in many cases the pseudoeffective and nef cones of the projectivised cotangent bundle of a smooth projective K3 surface. We then use these results to construct explicit families of smooth curves on which the restriction of the cotangent bundle is not semistable (and hence not nef). In particular, this leads to a counterexample to a question of Campana–Peternell.

• Ottem, John Christian (2020). Nef cycles on some hyperkähler fourfolds. arXiv.org. ISSN 2331-8422. Full text in Research Archive
• Ottem, John Christian & Schreieder, Stefan (2019). On deformations of quintic and septic hypersurfaces. Journal des Mathématiques Pures et Appliquées. ISSN 0021-7824. 135. doi: 10.1016/j.matpur.2019.12.013. Full text in Research Archive Show summary

  An old question of Mori asks whether in dimension at least three, any smooth specialization of a hypersurface of prime degree is again a hypersurface. A positive answer to this question is only known in degrees two and three. In this paper, we settle the case of quintic hypersurfaces (in arbitrary dimension) as well as the case of septics in dimension three. Our results follow from numerical characterizations of the corresponding hypersurfaces. In the case of quintics, this extends famous work of Horikawa who analysed deformations of quintic surfaces.

• Mongardi, Giovanni & Ottem, John Christian (2019). Curve classes on irreducible holomorphic symplectic varieties. arXiv.org. ISSN 2331-8422. doi: 10.1142/S0219199719500780. Full text in Research Archive
• Mongardi, Giovanni & Ottem, John Christian (2019). Curve classes on irreducible holomorphic symplectic varieties. Communications in Contemporary Mathematics. ISSN 0219-1997. 22(7). doi: 10.1142/S0219199719500780. Full text in Research Archive Show summary

  We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3[n]-type and of generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.

• Ottem, John Christian & Rennemo, Jørgen Vold (2018). A counterexample to the birational Torelli problem for Calabi-Yau 3-folds. Journal of the London Mathematical Society. ISSN 0024-6107. 97(3), p. 427–440. doi: 10.1112/jlms.12111. Full text in Research Archive Show summary

  The Grassmannian Gr(2,5) is embedded in ℙ9 via the Pl\"ucker embedding. The intersection of two general PGL(10)-translates of Gr(2,5) is a Calabi-Yau 3-fold X, and the intersection of the projective duals of the two translates is another Calabi-Yau 3-fold Y, deformation equivalent to X. Applying results of Kuznetsov and Jiang-Leung-Xie shows that X and Y are derived equivalent, which by a result of Addington implies that their third cohomology groups are isomorphic as polarised Hodge structures. We show that X and Y provide counterexamples to a certain "birational" Torelli statement for Calabi-Yau 3-folds, namely, they are deformation equivalent, derived equivalent, and have isomorphic Hodge structures, but they are not birational.

• Lehmann, Brian & Ottem, John Christian (2018). Positivity of the diagonal. Advances in Mathematics. ISSN 0001-8708. 335, p. 664–695. doi: 10.1016/j.aim.2018.07.002. Full text in Research Archive Show summary

  We study how the geometry of a projective variety X is reflected in the positivity properties of the diagonal ΔX considered as a cycle on X×X. We analyze when the diagonal is big, when it is nef, and when it is rigid. In each case, we give several implications for the geometric properties of X. For example, when the diagonal is big, we prove that the Hodge groups Hk,0(X) vanish for k>0. We also classify varieties of low dimension where the diagonal is nef and big.

• Ottem, John Christian & Schreieder, Stefan (2018). On deformations of quintic and septic hypersurfaces. arXiv.org. ISSN 2331-8422. doi: 10.1016/j.matpur.2019.12.013. Full text in Research Archive
• Ottem, John Christian & Gounelas, Frank (2018). Remarks on the positivity of the cotangent bundle of a K3 surface. arXiv.org. ISSN 2331-8422. Full text in Research Archive
• Lesieutre, John & Ottem, John Christian (2016). Curves disjoint from a nef divisor. The Michigan mathematical journal. ISSN 0026-2285. 65(2), p. 321–332. doi: 10.1307/mmj/1465329015. Full text in Research Archive Show summary

  On a projective surface it is well known that the set of curves orthogonal to a nef line bundle is either finite or uncountable. We show that this dichotomy fails in higher dimension by constructing an effective, nef line bundle on a threefold that is trivial on countably infinitely many curves. This answers a question of Totaro. As a pleasant corollary, we exhibit a quasi-projective variety with only a countably infinite set of complete, positive-dimensional subvarieties.

• Ottem, John Christian; Prendergast-Smith, Artie & Broomhead, Nathan (2016). Partially ample line bundles on toric varieties. Glasgow Mathematical Journal. ISSN 0017-0895. 58(3), p. 587–598. doi: 10.1017/S001708951500035X. Full text in Research Archive Show summary

  In this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone of q-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We prove a restriction theorem for big q-ample line bundles, and deduce that q-ampleness of the anticanonical bundle is not invariant under flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line bundles.

• Coskun, Izzet; Lesieutre, John & Ottem, John Christian (2016). Effective cones of cycles on blowups of projective space. Algebra & Number Theory. ISSN 1937-0652. 10(9), p. 1983–2014. doi: 10.2140/ant.2016.10.1983. Full text in Research Archive Show summary

  In this paper, we study the cones of higher codimension (pseudo)effective cycles on point blow-ups of projective space. We determine bounds on the number of points for which these cones are generated by the classes of linear cycles, and for which these cones are finitely generated. Surprisingly, we discover that for (very) general points, the higher codimension cones behave better than the cones of divisors. For example, for the blow-up Xnr of Pn, n>4, at r very general points, the cone of divisors is not finitely generated as soon as r>n+3, whereas the cone of curves is generated by the classes of lines if r<=2n. In fact, if Xnr is a Mori Dream Space then all the effective cones of cycles on Xnr are finitely generated.

• Ottem, John Christian (2016). On subvarieties with ample normal bundle. Journal of the European Mathematical Society (Print). ISSN 1435-9855. 18(11), p. 2459–2468. doi: 10.4171/JEMS/644. Full text in Research Archive Show summary

  We show that a pseudoeffective R-divisor has numerical dimension 0 if it is numerically trivial on a subvariety with ample normal bundle. This implies that the cycle class of a curve with ample normal bundle is big, which gives an affirmative answer to a conjecture of Peternell. We also give other positivity properties of such subvarieties.

• Ottem, John Christian (2015). Birational geometry of hypersurfaces in products of projective spaces. Mathematische Zeitschrift. ISSN 0025-5874. 280(1), p. 135–148. doi: 10.1007/s00209-015-1415-x. Full text in Research Archive
• Kleppe, Jan Oddvar & Ottem, John Christian (2015). Components of the Hilbert scheme of space curves on low-degree smooth surfaces. International Journal of Mathematics. ISSN 0129-167X. 26(2). doi: 10.1142/S0129167X15500172. Full text in Research Archive Show summary

  We study maximal families W of the Hilbert scheme, H(d,g)_sc, of smooth connected space curves whose general curve C lies on a smooth surface S of degree s. We give conditions on C under which W is a generically smooth component of H(d,g)_sc and we determine dim W. If s = 4 and W is an irreducible component of H(d;g)_sc, then the Picard number of S is at most 2 and we explicitly describe, also for s > 4, non-reduced and generically smooth components in the case Pic(S) is generated by the classes of a line and a smooth plane curve of degree s-1. For curves on smooth cubic surfaces the first author finds new classes of non-reduced components of H(d;g)_sc, thus making progress in proving a conjecture for such families.

• Ottem, John Christian (2013). Cox rings of K3 surfaces with Picard number 2. Journal of Pure and Applied Algebra. ISSN 0022-4049. 217(4), p. 709–715. doi: 10.1016/j.jpaa.2012.09.016. Full text in Research Archive Show summary

  We study presentations of Cox rings of K3 surfaces of Picard number 2. In particular we consider the Cox rings of classical examples of K3 surfaces, such as quartic surfaces containing a line and doubly elliptic K3 surfaces.

• Blekherman, Grigoriy; Hauenstein, Jonathan; Ottem, John Christian; Ranestad, Kristian & Sturmfels, Bernd (2012). Algebraic boundaries of Hilbert's SOS cones. Compositio Mathematica. ISSN 0010-437X. 148(6), p. 1717–1735. doi: 10.1112/S0010437X12000437. Full text in Research Archive Show summary

  We study the geometry underlying the difference between non-negative polynomials and sums of squares (SOS). The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether–Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized, respectively, by the Severi variety of plane rational sextics and by the variety of quartic symmetroids. This research was originally published in Compositio Mathematica. © Foundation Compositio Mathematica.

• Ottem, John Christian (2012). Ample subvarieties and q-ample divisors. Advances in Mathematics. ISSN 0001-8708. 229(5), p. 2868–2887. doi: 10.1016/j.aim.2012.02.001. Full text in Research Archive
• Ottem, John Christian (2011). ON THE COX RING OF P-2 BLOWN UP IN POINTS ON A LINE. Mathematica Scandinavica. ISSN 0025-5521. 109(1), p. 22–30. doi: 10.7146/math.scand.a-15175. Full text in Research Archive Show summary

  We show that the blow-up of P^2 in n points on a line has finitely generated Cox ring. We give explicit generators for the ring and calculate its defining ideal of relations.

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• Skogvoll, Vidar; Garvang, Even Sletteng; Brugård, Anne & Ottem, John Christian (2024). Under Kappa - #29 - The Riddler - Gode nøtter, smarte dyr og spillteori med Anne Brugård og John Christian Ottem. [Radio]. Podcast. Full text in Research Archive
• Brugård, Anne; Ottem, John Christian & Skogvoll, Vidar (2024). Abelegøyer - #42 - Påskespesial ft. Under Kappa. [Radio]. Podcast. Full text in Research Archive
• Ottem, John Christian (2021). Tropical degenerations and stable rationality. Full text in Research Archive
• Ottem, John Christian (2021). Enriques surface fibrations with even index. Full text in Research Archive
• Ottem, John Christian (2021). Tropical degenerations and stable rationality. Full text in Research Archive
• Ottem, John Christian (2021). Tropical degenerations and stable rationality. Full text in Research Archive
• Ottem, John Christian & Ottem, John Christian (2021). Enriques surface fibrations with even index. Full text in Research Archive
• Ottem, John Christian (2020). On (2,3)-fourfolds. Full text in Research Archive
• Ottem, John Christian (2020). Tropical degenerations and stable rationality. Full text in Research Archive
• Ottem, John Christian (2020). Enriques surface fibrations with non-algebraic integral Hodge classes. Full text in Research Archive
• Ottem, John Christian (2020). Tropical degenerations and stable rationality. Full text in Research Archive
• Ottem, John Christian (2019). Counterexamples to the integral Hodge conjecture. Full text in Research Archive
• Ottem, John Christian (2019). A counterexample to the birational Torelli problem. Full text in Research Archive
• Ottem, John Christian (2019). Deformations of hypersurfaces. Full text in Research Archive
• Ottem, John Christian (2019). Counterexamples to the integral Hodge conjecture. Full text in Research Archive
• Ottem, John Christian (2019). A pencil of Enriques surfaces with non-algebraic integral Hodge classes. Full text in Research Archive
• Ottem, John Christian (2019). A pencil of Enriques surfaces with non-algebraic integral Hodge classes. Full text in Research Archive
• Ottem, John Christian (2018). Failure of the Integral Hodge conjecture for 3-folds. Full text in Research Archive
• Ottem, John Christian (2018). The birational Torelli problem for Calabi-Yau 3-folds. Full text in Research Archive
• Ottem, John Christian (2018). Integral Hodge conjecture for threefolds. Full text in Research Archive
• Ottem, John Christian (2018). Integral hodge conjecture for holomorphic symplectic varieties. Full text in Research Archive
• Ottem, John Christian (2018). The integral Hodge conjecture for threefolds. Full text in Research Archive
• Ottem, John Christian (2018). Positivity of the higher codimension cycles. Full text in Research Archive
• Ottem, John Christian (2018). Integral Hodge conjecture for threefolds. Full text in Research Archive
• Ottem, John Christian (2017). Nef cycles on some hyperkahler manifolds. Full text in Research Archive
• Ottem, John Christian (2017). Nef cycles on some hyperkahler fourfolds. Full text in Research Archive
• Ottem, John Christian (2017). Semiample line bundles in positive characteristic. Full text in Research Archive
• Ottem, John Christian (2017). Nef cycles on some hyperkahler fourfolds. Full text in Research Archive
• Ottem, John Christian (2017). A counterexample to the birational Torelli problem for Calabi-Yau threefolds. Full text in Research Archive
• Ottem, John Christian (2017). Nef cycles of higher codimension. Full text in Research Archive
• Ottem, John Christian (2017). A counterexample to the birational Torelli problem for Calabi Yau threefolds. Full text in Research Archive
• Ottem, John Christian (2017). Positivity of the diagonal. Full text in Research Archive
• Ottem, John Christian (2017). Nef cycles on some hyperkahler manifolds. Full text in Research Archive
• Ottem, John Christian (2017). Positivity of the diagonal. Full text in Research Archive
• Ottem, John Christian (2017). The Birational Torelli problem for Calabi-Yau 3-folds. Full text in Research Archive
• Ottem, John Christian (2017). Positivity of the diagonal. Full text in Research Archive
• Ottem, John Christian (2016). Moduli of K3s via Global Torelli. Full text in Research Archive
• Ottem, John Christian (2016). Effective cones of cycles on blow-ups of projective space. Full text in Research Archive
• Ottem, John Christian (2016). Effective and nef cycles in higher codimension. Full text in Research Archive
• Ottem, John Christian (2016). Effective cycles on blow-ups of projective space. Full text in Research Archive
• Ottem, John Christian (2016). Effective cycles of higher codimension. Full text in Research Archive
• Ottem, John Christian (2016). Higher codimension cycles. Full text in Research Archive
• Ottem, John Christian (2016). K3 surfaces. Full text in Research Archive
• Ottem, John Christian (2016). Nef cycles on hyperkahler manifolds. Full text in Research Archive
• Ottem, John Christian (2016). Effective cones of cycles on blow-ups of projective space. Full text in Research Archive
• Ottem, John Christian (2016). Positivity in higher codimension. Full text in Research Archive
• Kleppe, Jan Oddvar & Ottem, John Christian (2012). Components of the Hilbert scheme of space curves on low-degree smooth surfaces. arXiv:1204.4819 [math.AG]. Full text in Research Archive Show summary

  We study maximal families W of the Hilbert scheme, H(d,g)_{sc}, of smooth connected space curves whose general curve C lies on a smooth surface S of degree s. For every integer s we give conditions on C under which W is a generically smooth component of H(d,g)_{sc} and we determine dim W. If s=4 and W is an irreducible component of H(d,g)_{sc}, then the Picard number of S is at most 2 and we explicitly describe non-reduced and generically smooth components in the case Pic(S) is generated by the classes of a line and a smooth plane cubic curve. For curves on smooth cubic surfaces we find new classes of non-reduced components of H(d,g)_{sc}, thus making progress in proving a conjecture for such families.

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