## Illinois Journal of Mathematics

### Cohomology of ideals in elliptic surface singularities

Tomohiro Okuma

#### Abstract

We introduce the the normal reduction number of two-dimensional normal singularities and prove that elliptic singularity has normal reduction number two. We also prove that for a two-dimensional normal singularity which is not rational, it is Gorenstein and its maximal ideal is a $p_{g}$-ideal if and only if it is a maximally elliptic singularity of degree $1$.

#### Article information

Source
Illinois J. Math., Volume 61, Number 3-4 (2017), 259-273.

Dates
Received: 12 October 2016
Revised: 13 January 2018
First available in Project Euclid: 22 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1534924827

Digital Object Identifier
doi:10.1215/ijm/1534924827

Mathematical Reviews number (MathSciNet)
MR3845721

Zentralblatt MATH identifier
06932504

#### Citation

Okuma, Tomohiro. Cohomology of ideals in elliptic surface singularities. Illinois J. Math. 61 (2017), no. 3-4, 259--273. doi:10.1215/ijm/1534924827. https://projecteuclid.org/euclid.ijm/1534924827

#### References

• M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129–136.
• K. Behnke and J. A. Christophersen, Hypersurface sections and obstructions (rational surface singularities), Compos. Math. 77 (1991), no. 3, 233–268.
• S. D. Cutkosky, A new characterization of rational surface singularities, Invent. Math. 102 (1990), no. 1, 157–177.
• S. Goto and Y. Shimoda, On the Rees algebras of Cohen-Macaulay local rings, Commutative algebra (Fairfax, Va., 1979), Lecture Notes in Pure and Appl. Math., vol. 68, Dekker, New York, 1982, pp. 201–231.
• S. Goto and K.-I. Watanabe, On graded rings. I, J. Math. Soc. Japan 30 (1978), no. 2, 179–213.
• C. Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), no. 2, 293–318.
• C. Huneke and I. Swanson, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006.
• S. Itoh, Integral closures of ideals generated by regular sequences, J. Algebra 117 (1988), no. 2, 390–401.
• M. Kato, Riemann-Roch theorem for strongly pseudoconvex manifolds of dimension $2$, Math. Ann. 222 (1976), no. 3, 243–250.
• H. B. Laufer, On minimally elliptic singularities, Amer. J. Math. 99 (1977), no. 6, 1257–1295.
• H. B. Laufer, On normal two-dimensional double point singularities, Israel J. Math. 31 (1978), no. 3–4, 315–334.
• J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Publ. Math. Inst. Hautes Études Sci. 36 (1969), 195–279.
• M. Morales, Calcul de quelques invariants des singularités de surface normale, Knots, braids and singularities (Plans-sur-Bex, 1982), Monogr. Enseign. Math., vol. 31, Enseignement Math., Geneva, 1983, pp. 191–203.
• M. Morales, Clôture intégrale d'idéaux et anneaux gradués Cohen-Macaulay, Géométrie algébrique et applications, I (La Rábida, 1984), Travaux en Cours, vol. 22, Hermann, Paris, 1987, pp. 151–171.
• A. Némethi, “Weakly” elliptic Gorenstein singularities of surfaces, Invent. Math. 137 (1999), no. 1, 145–167.
• T. Okuma, Numerical Gorenstein elliptic singularities, Math. Z. 249 (2005), no. 1, 31–62.
• T. Okuma, K.-I. Watanabe and K.-I. Yoshida, Good ideals and $p_g$-ideals in two-dimensional normal singularities, Manuscripta Math. 150 (2016), no. 3–4, 499–520.
• T. Okuma, K.-I. Watanabe and K.-I. Yoshida, Rees algebras and $p_g$-ideals in a two-dimensional normal local domain, Proc. Amer. Math. Soc. 145 (2017), no. 1, 39–47.
• T. Okuma, K.-I. Watanabe and K.-I. Yoshida, A characterization of two-dimensional rational singularities via Core of ideals, J. Algebra 499 (2018), 450–468.
• M. Reid, Chapters on algebraic surfaces, Complex algebraic geometry, IAS/Park City Math. Ser., vol. 3, Amer. Math. Soc., Providence, RI, 1997.
• A. Röhr, A vanishing theorem for line bundles on resolutions of surface singularities, Abh. Math. Semin. Univ. Hambg. 65 (1995), 215–223.
• J. D. Sally, Tangent cones at Gorenstein singularities, Compos. Math. 40 (1980), no. 2, 167–175.
• M. Tomari, A $p_g$-formula and elliptic singularities, Publ. Res. Inst. Math. Sci. 21 (1985), no. 2, 297–354.
• M. Tomari and K. Watanabe, Filtered rings, filtered blowing-ups and normal two-dimensional singularities with “star-shaped” resolution, Publ. Res. Inst. Math. Sci. 25 (1989), no. 5, 681–740.
• P. Wagreich, Elliptic singularities of surfaces, Amer. J. Math. 92 (1970), 419–454.
• S. S. T. Yau, Gorenstein singularities with geometric genus equal to two, Amer. J. Math. 101 (1979), no. 4, 813–854.
• S. S. T. Yau, Hypersurface weighted dual graphs of normal singularities of surfaces, Amer. J. Math. 101 (1979), no. 4, 761–812.
• S. S. T. Yau, On maximally elliptic singularities, Trans. Amer. Math. Soc. 257 (1980), no. 2, 269–329.