DOI QR코드

DOI QR Code

ON A COMPUTATION OF PLURIGENUS OF A CANONICAL THREEFOLD

  • Shin, Dong-Kwan (Department of Mathematics Konkuk University)
  • Received : 2015.03.05
  • Published : 2016.01.31

Abstract

For a canonical threefold X, it is known that $p_n$ does not vanish for a sufficiently large n, where $p_n=h^0(X,\mathcal{O}_X(nK_X))$. We have shown that $p_n$ does not vanish for at least one n in {6, 8, 10}. Assuming an additional condition $p_2{\geq}1$ or $p_3{\geq}1$, we have shown that $p_{12}{\geq}2$ and $p_n{\geq}2$ for $n{\geq}14$ with one possible exceptional case. We have also found some inequalities between ${\chi}(\mathcal{O}_X)$ and $K^3_X$.

Keywords

canonical threefold;threefold of general type;plurigenus

References

  1. J. A. Chen and M. Chen, Explicit birational geometry of threefolds of general type, I, Ann. Sci. Ec. Norm. Super. (4) 43 (2010), no. 3, 365-394. https://doi.org/10.24033/asens.2124
  2. J. A. Chen and M. Chen, Explicit birational geometry of 3-folds of general type, II, J. Differential Geom. 86 (2010), no. 2, 237-271. https://doi.org/10.4310/jdg/1299766788
  3. A. R. Fletcher, Contributions to Riemann-Roch on Projective 3-folds with Only Canon-ical Singularities and Applications, In: Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 221-231, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987.
  4. J. Kollar, Higher direct images of dualizing sheaves I, Ann. of Math. 123 (1986), no. 1, 11-42 https://doi.org/10.2307/1971351
  5. M. Reid, Young Person's guide to canonical singularities, In: Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 345-414, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987.
  6. D. Shin, On a computation of plurigenera of a canonical threefold, J. Algebra 309 (2007), no. 2, 559-568. https://doi.org/10.1016/j.jalgebra.2006.03.034
  7. J. Kollar, Higher direct images of dualizing sheaves II, Ann. of Math. 124 (1986), 171-202. https://doi.org/10.2307/1971390