# Home Page, Shigeru MUKAI

Japanese
Kakenhi Kiban (S) (2013-2017)

**List of publications ****
List of Publications (1995-2003)
**

**
RIMS preprint (2001-) ****
**** #1343: Conterexample to Hilbert's fourteenth problem for the 3-dimensional additive group, 2001
****#1372: Geometric realization of T-Shaped root systems and counterexamples to Hilbert's fourteenth problem, 2002. (Algebraic Transformation Group and Algebraic Varieties, pp.120-130, Springer Verlag, Berlin, 2004)
**** #1395: Curves and symmetric spaces, II, 2003. (Ann. of Math. 172(2010), 1539-1558.)
****#1403: Canonical curves of genus eight, 2003 (with Manabu IDE, Proc. Japan Acad. 79(2003), 59-64)
****#1422: Plane quartics and Fano threefolds of genus twelve, 2003. (The Fano Conference, pp.563-572, 2004, Univ. Torino)
****#1502: Finite generation of the Nagata invariant rings in A-D-E cases, 2005.
****#1505: Counterexample of Kodaira's vanishing and Yau's ineqauality in higher dimensional variety of characteristic p>0, 2005. (Originally typeset around 1980.)
**** #1544: Numerically trivial involutions of Enriques surfaces, May 2006.
(One type was omitted in the classification in 1984.) (Kyoto J. Math. 50(2010), no. 4, 889-902.)
****#1562: (with H. Nasu) Obstructions to deforming curves on a 3-fold, I --- a generalization of Mumford's example and an application to Hom schemes ---, July 2006.
( Journal of Algebraic Geometry, 18(2009), no. 4, 691-709.)
**** #1633: Kummer's quartics and numerically reflective involutions of Enriques surfaces, June 2008. (Conjecture 3 has been solved by H. Ohashi in his recent preprint "Enriques surfaces covered by Jacobian Kummer surfaces".) (J. Math. Soc. Japan. 64(2012), 231-246.)
**** Addendum to #1544: Addendum to "Numerically trivial involutions of Enriques surfaces", October 2009.
(Involutions of Lieberman type are not cohomologically trivial.)
**** #1736: Counterexamples of Kodaira's vanishing and Yau's inequality in
positive characteristics, 2011. (Revised English translation of "On counterexamples of Kodaira's vanishing and Yau's inequality in
positive characteristics (in Japanese),
Proceeding of Kinosaki algebraic geometry symposium, 1979, pp. 9--31.)
**** #1743 : K3 surfaces of genus sixteen, February 2012.
**
Abstract. The generic polarized $K3$ surface (S, h) of genus 16,
that is, (h^2)=30, is described in a certain compactifeid moduli space \mathcal{T} of twisted cubics in P^3, as a complete intersection with respect to an almost homogeneous vector bundle of rank 10.
As corollary we prove the unirationality of the moduli space \mathcal{F}_{16} of such K3 surfaces.

** #1751 : (with H. Ohashi) Enriques surfaces of Hutchinson-Gopel type and Mathieu automorphism, June 2012. ("Arithmetic and Geometry of K3 surfaces and Calabi-Yau Threefoolds", eds. M. Schuett, R. Laza and N. Yui, Fields Inst. for Research in Math. Sciences, Springer-Verlag, 2013)
**

** #1795 : (with
H. Ohashi) The automorphism groups of Enriques surfaces covered by symmetric quartic surfaces, February 2014.
**

**
INI preprint (2011) ****
**** NI11008-MOS: Igusa quartic and Steiner surfaces, Contemp. Math. 564(2012), 205-210.
**
Abstract. The Igusa quartic has a morphism of degree 8 onto itself.
Via this self-morphism, the Satake compactification of the moduli of principally
polarized abelian surfaces with Goepel triples (as well as usual p.p.a.s.'s with full level-2 structures) is isomorphic to the Igusa quartic. We also determine the action of Fricke involution on the moduli.

** My talk at Kinosaki Symposium in October 2010 ****
**** Enriques surfaces and root systems -- Enriques surfaces of type E7 --
Kinosaki10Oct.pdf
**

** Abstract of my talk at Oberwolfach in September 2010 ****
**** Enriques surfaces with many (semi-)symplectic automorphisms
Oberwolfach10Sep.pdf
**

** Abstract of my talk at Oberwolfach in January 2010 ****
**** Polarized K3 surfaces of genus 16
Oberwolfach10Jan.pdf
**

** Abstract of my talk at Oberwolfach in September 2009 ****
**** Numerically reflective involutions of Enriques surfaces
Oberwolfach09Sep.pdf
**

** Polarized K3 surfaces of genus thirteen, in "Moduli spaces and arithmetic geometry", Adv. Stud. Pure Math. 45, Math. Soc. Japan, Tokyo, 2006, pp. 315--326.
**** genus13.pdf
**

** Abstract of my talk at Oberwolfach in June 2006 ****
**** A generalization of Mumford's example (joint work with H. Nasu)
owr06.ps
owr06.pdf
**

** Abstracts of my talks at Oberwolfach in July 2004 and in February 2005****
**** Finite and infinite generation of Nagata invariant ring
Oberwolfach04.ps
Oberwolfach04.pdf
**** Geometric proof of finite generation of certain rings of invariants
Oberwolfach05.ps
Oberwolfach05.pdf
**

** Vector bundles on a K3 surface ****
**** math.AG/0304303
**

** Moduli of abelian surfaces, and regular polyhedral group ****
**** polyhedral.pdf
**

**Warwick Preprint:1998****
****#13: Abelian variety and spin representation
warwick13.ps
warwick13.pdf
****#14: Equations defining a space curve
warwick14.ps
warwick14.pdf
****#15: Simple Lie algebras and Legendre variety
warwick15.ps
warwick15.pdf
**

** Non-abelian Brill-Noether theory and Fano 3-folds****
**** alg-geom/9704015
**

** Vector bundles and Brill-Noether theory, in "Current Topics in Complex Algebraic Geometry", Cambridge Univ. Press, 1995 ****
**** MSRI.pdf
**

** Curves and Grassmannians, 1992, Inchoen, Korea (Algebraic Geometry and related Topics, pp.19-40, International Press, 1993, Cambridge, MA) ****
****Inchoen.ps
**

** Fano 3-folds, in "Complex projective geometry", Cambridge Univ. Press, 1992, pp. 255--263.****
**** Trieste.pdf
**

** Polarized K3 surfaces of genus 18 and 20, in "Complex projective geometry", Cambridge Univ. Press, 1992, pp. 264--276.
**** Bergen.pdf
**

**
Problems on characterization of the complex projective space, in `Birational Geometry of Algebraic Varieties, Open Problems, Kataata, 1988' (the 23rd Int'l Symp., Taniguchi Foundation), 1988, pp. 57--60.
**** Taniguchi1988.pdf
****
**

** On the moduli spaces of bundles on K3 surfaces, I, "Vector bundles on Algebraic Varieties", Tata Institute of Fundamental Research, Bombay, January, 1984 ****
**** Tata.pdf
**

** Fourier functor and its application to the moduli of bundles on an abelian variety, in `Algebraic Geometry, Sendai, 1985', Series : Adv. Stud. Pure Math., vol. 10, (T. Oda ed.), 1987, pp. 515-550. ****
**** Fourier1987.pdf
**

** (Shigefumi Mori, ---), Classification of Fano 3-folds with B_2 >= 2, I, in `Algebraic and Topological Theories -- to the memory of Dr. Takehiko Miyata', (M. Nagata ed.), Kinokuniya, 1985, pp. 496-545. ****
**** Fano1985.pdf
**

** (Shigefumi Mori, ---), On Fano 3-folds with B_2 >= 2, in `Algebraic Varieties and Analytic Varieties', Series : Adv. Stud. Pure Math., vol. 1, (S. Iitaka ed.), 1983, pp. 101-129.
****
**** Fano1983.pdf
**

**
Errata of "An Introduction to Invariants and Moduli"
(Cambridge Univ. Press 2003) **

** errata.ps
errata.pdf
**
14) page 24, line 8 The coefficient "3" of the middle term in the right hand side should read "6".

Last modified: May 14, 2014.