# 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.)
KURI-Repository
****#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.)
KURI-Repository
**** 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.)
KURI-Repository
**** #1743 : K3 surfaces of genus sixteen, February 2012.
"Minimal models and extremal rays", Adv. Stud. Pure Math. 70, Math. Soc. Japan, Tokyo, 2016, pp. 379--396. genus16.pdf
**
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.
In ``Recent Advances in Algebraic Geometry", A volume in honor of Rob Lazarsfeld's 60th birthday, eds. Hacon, Mustata and Popa, Cambridge Univ. Press, 2015, 307--320.
KURI-Repository
**

**
****INI preprint
(Programme: Moduli spaces MOS)
NI11008-MOS: Igusa quartic and Steiner surfaces ****
**
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.

** Igusa quartic and Steiner surfaces, Contemp. Math. 564(2012), 205-210.
Igusa-Steiner.pdf
**** Correection published in "Siegel modular forms of genus 2 and level 2" by F. Clery, G. van der Geer and S. Grushevsky (Int. J. Math., 26(2015)) as an appendix.
correction.pdf
**

** My talk at Waseda Univ. in March 2022 (prelim. ver.)****
JE-symplectic
**** Vinberg-Conway chain terminates just after a supersingular symplectic 6-fold with 100 (-2) divisors.
** Abstract: Leech roots calculate fundamental domains of the hyperbolic lattices U+D_n up to n=22. Generalizing a result of Vinberg (1983, n=18), we apply this calculation for n < 21 to (semi-)automorphism groups of various K3 surfaces, and realize the case n=21 as the Picard lattice of the symplectic manifold in characteristic 2 in the title, which is expected to have an action of the Higman-Sims simple group.

** My talk at YMSC zoom seminar in Tsinghua Univ. (Dec. 2021)****
**** Prime Fano threefolds and Leech-like lattices
Fano+K3^ss1
**

** My talk at a zoom meeting in occasion of Prof. Mori's 70th birthday (Handscript, page 2 replaced)****
**** Uniruledness of M11, and prime Fano 3-folds V22 of genus 12
Genus11+poristic neighbors
**

** 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
**

** Geometric realization of root systems and the Jacobians of del Pezzo surfaces,
in "Complex geometry in Osaka: in honour of Professor Akira Fujiki on the occasion of his 60th birthday" (R. Goto, N. Honda, M. Ishida, Y. Namikawa and K. Yoshikawa, eds),
Osaka Math. Publ., Osaka University, 2008, pp. 134-136.****
**** Fujiki60.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, Proceedings of the ICM, Beijing 2002, vol. 2, 495-502.****
**** math/0304303
**

** Moduli of abelian surfaces, and regular polyhedral group (Update of Hokudai-bilevel_1999.pdf in Hannover 1999.) ****
**** 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
**

** New developments in the theory of Fano threefolds: vector bundle method and moduli problem, Sugaku Exposition 15(2002), English translation of Sugaku 47(1995) ****
**** Sugaku1995.pdf
**

** Duality of polarized K3 surfaces, in "New trends in algebraic geometry (1996 Warwick EuroConference)", CUP, 1999, pp. 311-326 ****
**** Euroconf.ps
**

** Curves and K3 surfaces of genus eleven, in "Moduli of vector bundles" (ed. M. Maruyama), Marcel Dekker, New York, 1996, pp. 189-197 ****
**** genus11.pdf
**

** 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) ****
****inha.pdf
**

** 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
****
**

** Curves, K3 surfaces and Fano 3-folds of genus =< 10, "Algebraic Geometry and Commutative Algebra in Honor of Masayoshi NAGATA, I", (H. Hijikata, H. Hironaka, M. Maruyama, H. Matsumura, M. Miyanishi, T. Oda, K. Ueno eds.), 1987, pp. 357-377, Kinokuniya, Tokyo. ****
**** genus_10.pdf
**

** On the moduli spaces of bundles on K3 surfaces, I, "Vector bundles on Algebraic Varieties", Tata Institute of Fundamental Research/ Oxford Univ. Press, 1987, pp. 341-413. ****
**** Tata.pdf
**

** Moduli of vector bundles on K3 surfaces, and symplectic manifolds, Sugaku Exposition 1(1987), English translation of Sugaku 39(1987) ****
**** Sugaku1987.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: March 14, 2022.