# Prof. Dr. Gavril Farkas

Leiter Lehrstuhl Algebraische Geometrie
Humboldt-Universität zu Berlin
Institut für Mathematik
Rudower Chaussee 25, Raum 1.401
10099 Berlin
Tel: +49 (30) 2093-45442
e-mail: farkas-at-math.hu-berlin.de

Sekretariat: Désirée Dzomo Quevedo
Tel: +49 (30) 2093-5823
Fax: +49 (30) 2093-5853
e-mail: dzomoqud-at-math.hu-berlin.de

• Lehre

• Humboldt Algebraic Geometry Seminar Im Sommersemester 2021 findet das Forschungsseminar mittwochs von 16:00 bis 17:00 Uhr online im Zoom-Format statt.

Aktuelles/Upcoming events

• Brill-Noether Theory, HU Berlin, 5-7 November 2019

• Moduli in Budapest, a workshop at the Institute of Mathematics of the Hungarian Academy, Budapest 4-6 April 2019

• Rationality of algebraic varieties, a conference on Schiermonnikoog Island, 15-19 April 2019

• Varieties, Polyhedra, Computation
Thematic Einstein Semester on Algebraic Geometry in Berlin, Winter Semester 2019/20. Coorganised with Peter Buergisser and Christian Haase.

• Algebraic geometry activities Internationale Tagungen und Forschungsseminare

Editorial work/Herausgebertätigkeit

• Algebraic Geometry -Managing Editor

• Algebra & Number Theory

• Documenta Mathematica -the Journal of the German Mathematical Society (DMV)

• EMS Surveys in Mathematical Sciences

• Mathematica (Cluj) -a publication of the Romanian Academy

Research/Forschungsgebiet

My field of work is algebraic geometry. A significant part of my research is concerned with the study of the geometry and topology of various moduli spaces. Among other things, I am interested in moduli of curves, enumerative geometry, syzygies of algebraic varieties and commutative algebra aspects, vector bundles on curves, abelian varieties and theta functions, Prym varieties, K3 surfaces and Brill-Noether type problems.

Current Projects/Laufende Projekte

• ERC Advanced Grant SYZYGY: Syzgies, moduli and topological invariants of groups (2020-2024)

• DFG Sachbeihilfe Syzygien und Moduli (2019-2021)

• Berlin Mathematical School-Executive Board member

Publications/Veröffentlichungen

Hodge classes on the moduli space of $W(E_6)$-covers and the geometry of $A_6$ pdf
(with V. Alexeev, R. Donagi, E. Izadi and A. Ortega)

Alternate Catalan numbers and curves with triple ramification pdf
(with R. Moschetti, J.C. Naranjo and G.P. Pirola)
Annali della Scuola Normale Superiore di Pisa 22 (2021), 665-690

Quadric rank loci on moduli of curves and K3 surfaces pdf
(with R. Rimányi)
Annales Scientifiques de L'Ecole Normale Superieure 53(2020), 945-992.
Accompanying Macaulay file,
here.

The uniformization of the moduli space of principally polarized abelian 6-folds pdf
(with V. Alexeev, R. Donagi, E. Izadi and A. Ortega)
Journal für die reine und angewandte Mathematik 761 (2020), 163-217

Koszul modules and Green's Conjecture pdf
(with M. Aprodu, S. Papadima, C. Raicu and J. Weyman)
Inventiones Mathematicae 218 (2019), 657-720

Linear syzygies on curves with prescribed gonality pdf
(with M. Kemeny)
Advances in Mathematics 356 (2019), 106810

The resolution of paracanonical curves of odd genus pdf
(with M. Kemeny)
Geometry & Topology 22 (2018), 4235-4257

Mercat's Conjecture for stable rank 2 vector bundles on generic curves pdf
(with B. Bakker)
American Journal of Mathematics 140 (2018), 1277-1295

The moduli space of twisted canonical differentials pdf
(with R. Pandharipande)
Journal of the Institute of Mathematics of Jussieu 17 (2018), 615-672

The universal K3 surface of genus 14 via cubic fourfolds pdf
(with A. Verra)
Journal de Mathématiques Pures et Appliquées 111 (2018), 1-20

Syzygies of Prym and paracanonical curves of genus 8 pdf
(with E. Colombo, A. Verra and C. Voisin)
Épijournal de Géométrie Algébrique 1 (2017)

Progress on syzygies of algebraic curves pdf
Lecture Notes of the Unione Matematica Italiana 21 (Moduli of Curves, Guanajuato 2016) 107-138

Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces pdf
(with M. Aprodu and A. Ortega)
Journal für die reine und angewandte Mathematik 730 (2017), 225-249

The Prym-Green conjecture for torsion line bundles of high order pdf
(with M. Kemeny)
Duke Mathematical Journal 166 (2017), 1103-1124

Singularities of the moduli space of level curves pdf
(with A. Chiodo)
Journal of the European Mathematical Society 19 (2017), 603-658

Du Val curves and the pointed Brill-Noether theorem pdf
(with N. Tarasca)
Selecta Mathematica 23 (2017), 2243-2259

Explicit Brill-Noether-Petri general curves pdf
(with E. Arbarello, A. Bruno and G. Sacca)
Commentarii Mathematici Helvetici 91(2016), 477-491

The universal abelian variety over $A_5$ dvi pdf
(with A. Verra)
Annales Scientifiques de L'Ecole Normale Superieure 49(2016), 521-543

Prym varieties and moduli of polarized Nikulin surfaces pdf
(with A. Verra)
Advances in Mathematics 290 (2016), 314-328

The generic Green-Lazarsfeld Conjecture pdf
(with M. Kemeny)
Inventiones Mathematicae 203 (2016), 265-301

Pointed Castelnuovo numbers pdf
(with N. Tarasca) Mathematical Research Letters 23(2016), 389-404

Restricted Lazarsfeld-Mukai bundles and canonical curves pdf
(with M. Aprodu and A. Ortega), Development of Moduli Theory (Kyoto 2013)
Advanced Studies in Pure Mathematics 69 (2016), 303-322.

The geometry of the moduli space of odd spin curves dvi pdf
(with A. Verra)
Annals of Mathematics 180 (2014), 927-970

Singularities of theta divisors and the geometry of $A_5$ dvi pdf
(with S. Grushevsky, R. Salvati Manni and A. Verra)
Journal of the European Mathematical Society 16 (2014), 1817-1848

Syzygies of torsion bundles and the geometry of the level $\ell$ modular variety over $\overline{\mathcal{M}}_g$ dvi pdf
(with A. Chiodo, D. Eisenbud and F.-O. Schreyer)
Inventiones Mathematicae 194 (2013), 73-118

The universal theta divisor over the moduli space of curves dvi pdf
(with A. Verra)
Journal de Mathématiques Pures et Appliquées 100 (2013), 591-605

The classification of universal Jacobians over the moduli space of curves dvi pdf
(with A. Verra)
Commentarii Mathematici Helvetici 88 (2013), 587-611

Brill-Noether with ramification at unassigned points dvi pdf
Journal of Pure and Applied Algebra 217(2013), 1838-1843

Effective divisors on moduli spaces of curves and abelian varieties pdf
(with D. Chen and I. Morrison) A Celebration of Algebraic Geometry 131-169
Clay Mathematics Proceedings, B. Hassett, J. McKernan, J. Starr, R. Vakil editors, Volume 18, 2013

Higher rank Brill-Noether theory on sections of K3 surfaces dvi pdf
(with A. Ortega)
International Journal of Mathematics 23 (2012) 1250075

Moduli of theta characteristics via Nikulin surfaces dvi pdf
(with A. Verra)
Mathematische Annalen 354 (2012), 465-496

Theta characteristics and their moduli dvi pdf
Milan Journal of Mathematics 80 (2012), 1-24

Prym varieties and their moduli pdf
Contributions to Algebraic Geometry-IMPANGA Lecture Notes, 215-257
P. Pragacz editor, EMS Congress Reports 2012

Green's Conjecture for general covers pdf
(with M. Aprodu) Compact moduli spaces and vector bundles-Athens, Georgia 2010
Contemporary Mathematics Vol. 564, 2012, 211-226

Brill-Noether geometry on moduli spaces of spin curves dvi pdf
Classification of algebraic varieties-Schiermonnikoog 2009, 259-276
C. Faber, G. van der Geer, E. Looijenga editors, EMS Congress Reports 2011

Koszul cohomology and applications to moduli dvi pdf
(with M. Aprodu)
Grassmannians, moduli spaces and vector bundles
Clay Mathematics Proceedings, E. Previato editor, Volume 14, 2011, 25-50

The Green Conjecture for smooth curves lying on arbitrary K3 surfaces dvi pdf
(with M. Aprodu)
Compositio Mathematica 147 (2011), 839-851

The maximal rank conjecture and rank two Brill-Noether theory dvi pdf
(with A. Ortega)
Pure and Applied Mathematics Quarterly 7 (2011), 1265-1296

Aspects of the birational geometry of $\mathcal{M}_g$ dvi pdf
"Geometry of Riemann surfaces and their moduli spaces"
Surveys in Differential Geometry Vol. 14 (2010), 57-111

The birational type of the moduli space of even spin curves dvi pdf
Advances in Mathematics 223 (2010), 433-443

Rational maps between moduli spaces of curves and Gieseker-Petri divisors dvi pdf
Journal of Algebraic Geometry 19 (2010), 243-284

The Kodaira dimension of the moduli space of Prym varieties dvi pdf
(with K. Ludwig) Journal of the European Mathematical Society 12 (2010), 755-795

The Fermat cubic and special Hurwitz loci in $\overline{\mathcal{M}}_g$ dvi pdf
Bulletin of the Belgian Mathematical Society "Simon Stevin" 5 (2009), 831-851
Special volume dedicated to the conference "Linear series and subschemes"-Ghent 2007

The global geometry of the moduli space of curves dvi pdf
Algebraic Geometry Seattle 2005, Proceedings of Symposia in Pure Mathematics 80 (2009), 125-147

Koszul divisors on moduli spaces of curves dvi pdf
American Journal of Mathematics 131 (2009), 819-867

The intermediate type of certain moduli spaces of curves dvi pdf
(with A. Verra), preprint from 2009

Higher ramification and varieties of secant divisors on the generic curve dvi pdf
Journal of the London Mathematical Society 78 (2008), 418-440

Syzygies of curves and the effective cone of $\overline{\mathcal{M}}_g$ dvi pdf
Duke Mathematical Journal 135(2006), 53-98

Gaussian maps, Gieseker-Petri loci and large theta-characteristics dvi pdf
J. reine angewandte Mathematik 581 (2005), 151-174

Effective divisors on $\overline{\mathcal{M}}_g$, curves on K3 surfaces and the Slope Conjecture dvi pdf
(with M. Popa)
Journal of Algebraic Geometry 14 (2005), 241-267

Regular components of moduli spaces of stable maps dvi pdf
Proceedings of the American Mathematical Society 131 (2003), 2027-2036

The Mori cones of moduli spaces of pointed curves of small genus dvi pdf
(with A. Gibney), Transactions of the American Mathematical Society 355 (2003), 1183-1199

Divisors on $\mathcal{M}_{g,g+1}$ and the minimal resolution conjecture for points on canonical curves dvi pdf
(with M. Mustata and M. Popa), Annales Scientifiques de L'Ecole Normale Superieure 36 (2003), 553-581

Brill-Noether loci and the gonality stratification of $\mathcal{M}_g$ dvi pdf
J. reine angewandte Mathematik 539 (2001), 185-200

The geometry of the moduli space of curves of genus 23 dvi pdf
Mathematische Annalen 318 (2000), 43-65

Effective divisors on M_g and a counterexample to the Slope Conjecture dvi pdf
(with M. Popa)

The effective cone of M_g and the Slope Conjecture dvi pdf
Max Planck Institute preprint (2003)
This is an extended version of my talk at the Mathematische Arbeitstagung 2003

Notes on limit linear series for vector bundles on curves dvi pdf
(with M. Popa), unpublished preprint, 2002

Lectures on moduli of curves
Proceedings of the Korean Institute for Mathematical Sciences, Volume 2, No. 1 (2007), 1-20

The birational geometry of the moduli space of curves pdf
Thesis, Universiteit van Amsterdam (2000), book (85 pages), available upon request

A multiplicity theorem in equivariant setting
(with Cs. Varga) Mathematica 38 (1996), 221-226

Lusternik-Schnirelmann theory on closed subsets of C^1-manifolds
(with Cs. Varga) Studia Univ. Babes-Bolyai Math. 38 (1993), no. 2, 75-89

On the completeness of Riemanniann manifolds
(with Cs. Varga) Studia Univ. Babes-Bolyai Math. 37 (1992), no. 4, 96-101

My papers reviewed on MathSciNet

PhD Students/Meine Doktoranden

Paul Larsen (PhD 2010), Applied Mori theory of the moduli space of stable pointed rational curves
Nicola Tarasca (PhD 2012), Geometric cycles on the moduli space of curves
Margherita Lelli-Chiesa (PhD 2012), Gieseker-Petri divisors and Brill-Noether theory of K3 sections-Mario Baldassari Prize 2015
Fabian Müller (PhD 2013), Effective divisors on moduli spaces of pointed stable curves
Gregor Bruns (PhD 2016), Divisors on moduli spaces of level curves
Emre Sertöz (PhD 2017), Enumerative geometry of double spin curves
Irfan Kadiköylu (PhD 2018), Rank stratifications of spaces of quadrics and moduli of curves
Daniele Agostini (PhD 2018), On syzygies of algberaic varieties with applications to moduli
Ignacio Barros (PhD 2018), K3 surfaces and moduli of holomorphic differentials
Mara Ungureanu (PhD 2018), Enumerative formulas of de Jonquieres type on algebraic curves
Irene Schwarz (PhD 2020), Birational geometry of moduli spaces of pointed curves

Current PhD students: Andrei Bud (2020-), Carlos Maestro Perez (2018-), Federico Moretti (2020-)