## Research

**Research Areas:**

**Research Interests:**

Degenerations and compact moduli spaces of algebraic varieties, including curves, surfaces, abelian varieties, other varieties with group action. Birational geometry and Minimal Model Program. Singularities appearing in MMP. Toric and spherical varieties. Derived categories. Extremal metrics.

## Selected Publications

**Selected Publications:**

## Papers and preprints

(with Hülya Argüz and Pierrick Bousseau) The KSBA moduli space of stable log Calabi-Yau surfaces

(with Philip Engel, D. Zack Garza, Luca Schaffler) Compact moduli of Enriques surfaces with a numerical polarization of degree 2

(with Xiaoyan Hu) Explicit compactifications of moduli spaces of secondary Burniat surfaces

(with Rita Pardini) Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces

Reflective hyperbolic 2-elementary lattices, K3 surfaces and hyperkahler manifolds, LMS Lecture Notes, to appear.

(with Philip Engel) Compact moduli of K3 surfaces. Annals of Math, 198 , no.2, 727–789 (2023).

(with Philip Engel) Compactifications of moduli spaces of K3 surfaces with a nonsymplectic involution.

Root systems and hyperkahler varieties

(with Philip Engel and Changho Han) Compact moduli of K3 surfaces with a nonsymplectic automorphism, Trans. Amer. Math. Soc. Ser. B 11 (2024) 144-163.

(with Philip Engel) The flex divisor of a K3 surface, IMRN, Volume 2023, Issue 10, May 2023, Pages 8356–8370, https://doi.org/10.1093/imrn/rnac089

(with Ron Donagi, Gavril Farkas, Elham Izadi, Angela Ortega) Hodge classes on the moduli space of W(E_{6})-covers and the geometry of A_{6}, *Pure Appl. Math. Q.* 18 (2022), no. 4, 1211–1263.(Supplementary computational materials are here),

(with Adrian Brunyate and Philip Engel) Compactifications of moduli of elliptic K3 surfaces: stable pair and toroidal, Geometry & Topology 26-8 (2022), 3525--3588. DOI 10.2140/gt.2022.26.3525.

(with Philip Engel and Alan Thompson) Stable pair compactification of moduli of K3 surfaces of degree 2, J. Reine Angew. Math.799(2023), 1–56.

(with Alan Thompson) ADE surfaces and their moduli, J. Algebraic Geom. 30 (2021), no. 2, 331–405.

(with Wenfei Liu) Log surfaces of Picard rank one from four lines in the plane, Eur. J. Math. 5 (2019), no.3, 622-639.

(with Wenfei Liu) On accumulation points of volumes of log surfaces, Izv. RAN Seriya Mat. 83 (2019), no.4, 5-25.

(with Wenfei Liu) Open surfaces of small volume, Algebraic Geometry 6 (3) (2019) 312--327.

(with Alan Thompson) Modular compactification of moduli of K3 surfaces of degree 2, (old version, left here for a historical record).

(with Ron Donagi, Gavril Farkas, Elham Izadi, Angela Ortega) The uniformization of the moduli space of principally polarized abelian 6-folds, J. Reine Angew. Math. 761 (2020), 163–217, https://dx.doi.org/10.4310/PAMQ.2022.v18.n4.a1

Divisors on Burniat surfaces, Advanced Studies in Pure Mathematics (Mathematical Society of Japan), 69 (2016), pp. 287-302.

Moduli of weighted hyperplane arrangements, Advanced Cources in Mathematics - CRM Barcelona, Springer Basel, 2015.

(with Rita Pardini) On the existence of ramified abelian covers, Rend. Sem. Mat. Univ. Politec. Torino 71, 3-4 (2013), 307-315.

(with Dmitri Orlov) Derived categories of Burniat surfaces and exceptional collections, Math. Annalen, 357, issue 2 (2013), 743-759.

(with Alexander Borisov) On the log discrepancies in Mori contractions, Proc. Amer. Math. Soc., 142 (2014), no. 11, 3687-3694.

(with Christopher Hacon) Non-rational centers of log canonical singularities, J. Algebra, 369 (2012), 1-15.

(with Ryan Livingson, Joe Tenini, et al.) Extended Torelli map to the Igusa blowup in genus 6, 7, and 8. (long version here.), Experimental Mathematics, 21(2), 2012, 193-203.

(with Rita Pardini) Non-normal abelian covers, Compositio Mathematica, 148, no. 4 (2012), 1051-1084.

(with Adrian Brunyate) Extending Torelli map to toroidal compactifications of Siegel space, Inventiones Mathematicae, 188 (2012), issue 1, 175-196.

(with Angela Gibney and David Swinarski) Higher level sl(2) conformal blocks divisors on M_{0,n}, Proc. Edinburgh Math. Soc., 57 (2014), 7-30.

(with David Swinarski) Nef divisors on M_{0,n} from GIT, *Geometry and Arithmetic*, 2012, European Math. Soc., EMS Congress Reports, pp. 1-22.

Weighted grassmannians and stable hyperplane arrangements (updated 6/19/08)

*Higher-dimensional analogues of stable curves,*

Proceedings of Madrid ICM2006, 2 (2006), 515-536, Eur. Math. Soc. Pub. House.

(with Michael Guy) *Moduli of weighted stable maps and their gravitational descendants,* Journal of the Institute of Mathematics of Jussieu, 7 (2008), no.3, 425-456.

*Limits of stable pairs,* Pure and Applied Math Quaterly, 4 (2008), no.3, 767-783.

(with Christopher Hacon and Yujiro Kawamata) *Termination of (many) 4-dimensional log flips*

(successor to "A note on termination of 4-dimensional flips" below)

Inventiones Mathematicae, 168 (2007), 433-448.

(with Viacheslav V. Nikulin) *Del Pezzo and K3 surfaces,*

Memoirs of the Mathematical Society of Japan, 15 (2006), x+149pp.

(with Allen Knutson) *Complete moduli of branchvarieties,* Jour. f.d. Reine Angew. Mathematik (Crelle's Journal), 639 (2010), 29-71.

(with Michel Brion) *Stable spherical varieties and their moduli*

International Math. Research Papers, vol. 2006 (2006), Article ID 46293, 57 pages.

*Compactified jacobians and Torelli map*, Publ. RIMS, Kyoto Univ. 40 (2004), 1241-1265. (40-year RIMS anniversary issue). (Here is a local copy.)

*A note on termination of 4-dimensional flips* (6/8/2004)

(with Viacheslav V. Nikulin) *Classification of log del Pezzo surfaces of index <=2*

(with Michel Brion) *Toric degenerations of spherical varieties*, Selecta Mathematica 10 (2004), 1-26.

(with Ludmil Katzarkov) *On K-stability of reductive varieties*, Geometric and Functional Analysis GAFA 15:2 (2005), 297-310.

(with Michel Brion) *Moduli of affine schemes with reductive group action*

Journal of Algebraic Geometry 14 (2005), 83-117.

(with Michel Brion) *Boundedness of spherical Fano varieties*, Proceedings of "The Fano conference" Turin Sept. 2002, pp. 69-80 (2004).

(with Michel Brion) *Stable reductive varieties I: Affine varieties*, Inventiones Math. 157/2 (2004), 224-274.

(with Michel Brion) *Stable reductive varieties II: Projective case*, Advances of Math. 184/2 (2004), 380-408.

*Complete moduli in the presence of semiabelian group action*, Annals of Math. 155 (2002), 611-708.

(with Christina Birkenhake and Klaus Hulek) *Degenerations of Prym varieties*, Jour. f.d. Reine Angew. Mathematik (Crelle's Journal) 553 (2002), 73-116.

(with Shigefumi Mori) *Bounding singular surfaces of general type*

In "Algebra, Arithmetic and Geometry with Applications", ed. by Christensen et al, Springer-Verlag (2003), 143-174.

(with Iku Nakamura) On Mumford's construction of degenerating abelian varieties, *Tohoku Mathematical Journal* 51(1999), 399-420.

*On extra components in the functorial compactification of A _{g}*, Moduli of Abelian Varieties (Texel island) (K.Faber, G. van der Geer, and F.Oort, eds.), Progress in Mathematics 195, Birkhauser, Boston, 2001, 1-9.

Log canonical singularities and complete moduli of stable pairs, arXiv alg-geom/9608013.

Compactified jacobians, arXiv alg-geom/9608012.

Moduli spaces M_{g,n}(W) for surfaces, Proceedings of the international conference held in Trento, Italy, June 1994. Walter de Gruyter, 1996.

On general elephants of Q-Fano 3-folds, *Compositio Math.* 90 (1994), 91-116.

Boundedness and K^{2} for log surfaces, *Internat. J. Math.* 5 (1994), no. 6., 779-810.

Two two-dimensional terminations, *Duke Math. J.* 69 (1993), no.3, 527-545.

Classification of log canonical surface singularities: Arithmetical proof. In: Flips and abundance for algebraic threefolds, *Asterisque*, vol. 211, 1992, 47-58.

Ample Weil divisors on K3 surfaces with Du Val singularities, *Duke Math. J.* 64 (1991), no.3., 617-624.

Theorems about good divisors on log Fano varieties (case of index r > n-2), Algebraic Geometry, Chicago (1989), *Lecture Notes in Math.*, vol 1479, Springer-Verlag, 1991, 1-9.

Fractional indices of log del Pezzo surfaces, *Math. USSR-Izv.*, 33 (1989), 613-629. Correction.

(with V.V. Nikulin) Classification of del Pezzo surfaces with log terminal singularities of index ≤2 and involutions on K3 surfaces, *Soviet Math. Dokl.* 39 (1989), 525-528.

(with I. Kolpakov) On quotient surfaces P^{2} by a finite group, *Uspekhi Mat. Nauk* 43 (1988), no.5, 171-172.

(with V.V. Niulin) Classification of del Pezzo surfaces with log terminal singularities of index ≤2, involutions on K3 surfaces and reflection groups in Lobachevsky spaces, *Reports in Mathematics and its Applications* 2 (1988), no.2, 51-150.

Rationality conditions for three-dimensional varieties with a pencil of del Pezzo surfaces of degree 4, *Math. Notes* 41 (1987), 408-411.

## Books edited

(with Arnaud Beauville, C. Herbert Clemens, Elham Izadi) Curves and Abelian Varieties: International conference March 30-April 2, 2007, University of Georgia, Athens. *Contemporary Mathematics,* American Math. Soc., Vol. 465 (2008), 270 pp.

(with Angela Gibney, Elham Izadi, Eduard Looijenga, János Kollár) Compact Moduli and Vector Bundles. Proceedings of the 2010 conference at the University of Georgia, Athens. *Contemporary Mathematics*, American Math. Soc., 564 (2012), 249 pp.

## Other Information

**Of note:**

## Some conferences I coorganized

April 28-30, 2023: Georgia Algebraic Geometry Symposium 2023, Athens, GA

March 3-5, 2017: Georgia Algebraic Geometry Symposium 2017, Athens, GA

October 17-19, 2014: Georgia Algebraic Geometry Symposium 2014, Athens, GA

October 18-20, 2013: Georgia Algebraic Geometry Symposium 2013, Athens, GA

May 21 - June 1, 2012: UGA school and conference on algebraic geometry, Athens, GA

October 21-24, 2010: Compact moduli and vector bundles, University of Georgia, Athens, GA

March 30 - April 3, 2007: Curves and abelian varieties, University of Georgia, Athens, GA

## Teaching

**Teaching:**

## My students

Vitaly Vologodsky, PhD 2003, *The extended Jacobi and Prym maps*

Tawanda Gwena, PhD 2004, *Degenerations of Prym varieties and Cubic threefold*

Michael Guy, PhD 2007, *Moduli of Weighted Stable Maps and Their Gravitational Descendants*

Peter Petrov, PhD 2007, *Nash problem on spaces of arcs*

Joe Rusinko, PhD 2007, *Equivalence of Mirror Families Constructed from Toric Degenerations of Flag Varieties*

Jaeho Shin, PhD 2013, *The reduction map for the moduli spaces of weighted hyperplane arrangements*

Xiaoyan (Shannon) Hu, PhD 2014, *The compactifications of moduli spaces of Burniat surfaces with 2≤K ^{2}≤5*

Joe Tenini, PhD 2014, *Results on an Extended Torelli Map and Singularities of Degenerate Abelian Varieties*

Adrian Brunyate, PhD 2015, *A Modular Compactification of the Space of Elliptic K3 Surfaces*

Luca Schaffler, PhD 2017, *The KSBA compactification of a 4-dimensional family of polarized Enriques surfaces*

Xian Wu, PhD 2019, *Stable Pair Compactification of the Moduli Space of Two Special Families of Calabi-yau 3-folds and Chow Quotients of Grassmannians by Diagonal Subtori*

Nolan Schock, PhD 20221, *Geometry of tropical compactifications of moduli spaces*