John Christian Ottem

• Ottem, John Christian & Lehmann, Brian (2018). Positivity of the diagonal. Advances in Mathematics.  ISSN 0001-8708.  335, s 665- 695 . doi: 10.1016/j.aim.2018.07.002 Fulltekst i vitenarkiv. Vis sammendrag

  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 & 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), s 427- 440 . doi: 10.1112/jlms.12111 Fulltekst i vitenarkiv. Vis sammendrag

  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.

• Ottem, John Christian (2017). Nef cycles on some hyperkahler fourfolds. arXiv.org.  ISSN 2331-8422. Fulltekst i vitenarkiv. Vis sammendrag

  We study the cones of surfaces on varieties of lines on cubic fourfolds and Hilbert schemes of points on K3 surfaces. From this we obtain new examples of nef cycles which fail to be pseudoeffective.

• 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), s 1983- 2014 . doi: 10.2140/ant.2016.10.1983 Vis sammendrag

  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.

• Lesieutre, John & Ottem, John Christian (2016). Curves disjoint from a nef divisor. The Michigan mathematical journal.  ISSN 0026-2285.  65(2), s 321- 332 . doi: 10.1307/mmj/1465329015 Vis sammendrag

  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 (2016). On subvarieties with ample normal bundle. Journal of the European Mathematical Society (Print).  ISSN 1435-9855.  18(11), s 2459- 2468 . doi: 10.4171/JEMS/644 Fulltekst i vitenarkiv. Vis sammendrag

  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; Prendergast-Smith, Artie & Broomhead, Nathan (2016). Partially ample line bundles on toric varieties. Glasgow Mathematical Journal.  ISSN 0017-0895.  58(3), s 587- 598 . doi: 10.1017/S001708951500035X Vis sammendrag

  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.

• 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 Fulltekst i vitenarkiv. Vis sammendrag

  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 (2015). Birational geometry of hypersurfaces in products of projective spaces. Mathematische Zeitschrift.  ISSN 0025-5874.  280(1), s 135- 148 . doi: 10.1007/s00209-015-1415-x
• Ottem, John Christian (2013). Cox rings of K3 surfaces with Picard number 2. Journal of Pure and Applied Algebra.  ISSN 0022-4049.  217(4), s 709- 715 . doi: 10.1016/j.jpaa.2012.09.016 Fulltekst i vitenarkiv.
• 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), s 1717- 1735 . doi: 10.1112/S0010437X12000437 Fulltekst i vitenarkiv.
• Ottem, John Christian (2012). Ample subvarieties and q-ample divisors. Advances in Mathematics.  ISSN 0001-8708.  229(5), s 2868- 2887 . doi: 10.1016/j.aim.2012.02.001
• 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), s 22- 30 Fulltekst i vitenarkiv. Vis sammendrag

  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.

Se alle arbeider i Cristin

• Ottem, John Christian (2018). The birational Torelli problem for Calabi-Yau 3-folds.
• Ottem, John Christian (2017). Nef cycles on some hyperkahler fourfolds.
• Ottem, John Christian (2017). Nef cycles on some hyperkahler fourfolds.
• Ottem, John Christian (2017). Nef cycles on some hyperkahler manifolds.
• Ottem, John Christian (2017). Nef cycles on some hyperkahler manifolds.
• Ottem, John Christian (2017). Positivity of the diagonal.
• Ottem, John Christian (2017). Positivity of the diagonal.
• Ottem, John Christian (2017). Positivity of the diagonal.
• Ottem, John Christian (2017). Semiample line bundles in positive characteristic.
• Ottem, John Christian (2017). The Birational Torelli problem for Calabi-Yau 3-folds.
• Ottem, John Christian (2016). Effective and nef cycles in higher codimension.
• Ottem, John Christian (2016). Effective cones of cycles on blow-ups of projective space.
• Ottem, John Christian (2016). Effective cones of cycles on blow-ups of projective space.
• Ottem, John Christian (2016). Effective cycles of higher codimension.
• Ottem, John Christian (2016). Effective cycles on blow-ups of projective space.
• Ottem, John Christian (2016). Higher codimension cycles.
• Ottem, John Christian (2016). K3 surfaces.
• Ottem, John Christian (2016). Moduli of K3s via Global Torelli.
• Ottem, John Christian (2016). Nef cycles on hyperkahler manifolds.
• Ottem, John Christian (2016). Positivity in higher codimension.
• Kleppe, Jan Oddvar & Ottem, John Christian (2012). Components of the Hilbert scheme of space curves on low-degree smooth surfaces. Vis sammendrag

  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.

Se alle arbeider i Cristin