Prof. Dr. Gavril Farkas


Direktor des Instituts für Mathematik
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) 2093-45440
Fax: +49 (30) 2093-5853
e-mail: dzomoqud-at-math.hu-berlin.de


  • Lehre

  • Humboldt Algebraic Geometry Seminar Im Wintersemester 2023 findet das Forschungsseminar mittwochs von 13:15 bis 14:45 Uhr statt.




    Aktuelles/Upcoming events

  • Curves, abelian varieties and their moduli, HU Berlin, 30-31 May 2024

  • Moduli spaces and modular forms, Schiermonnikoog Island, 15-19 April 2024

  • Hilbert scheme of points, HU Berlin, 28 September 2023

  • Northern German Algebraic Geometry Seminar and the award of the Algebraic Geometry Prize and the Compositio Prize, HU Berlin 9-10 February 2023

  • Resonance varieties, topological invariants of groups, moduli, HU Berlin, 15-18 November 2022

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

  • 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, geometric group theory and resonance varieties.

    Current Projects/Laufende Projekte

  • ERC Advanced Grant SYZYGY: Syzgies, moduli and topological invariants of groups (2020-2026)
    (see also here)

  • DFG Sachbeihilfe Syzygien und Moduli (2019-2023)

  • Berlin Mathematical School-Cochair (2022-)

    Curriculum Vitae: dvi pdf


    Publications/Veröffentlichungen

    The uniruledness of the Prym moduli space of genus 9 pdf
    (with A. Verra)

    On the Kodaira dimension of $\overline{\mathcal{M}}_{16}$ pdf
    (with A. Verra)

    Reduced resonance schemes and Chen ranks pdf
    (with M. Aprodu, C. Raicu and A. Suciu)

    The Kodaira dimension of $\overline{\mathcal{M}}_{22}$ and $\overline{\mathcal{M}}_{23}$ pdf
    (with D. Jensen and S. Payne)
    Notes of my talk on this topic given at ETH in June 2020, to be found
    here

    Higher resonance schemes and Koszul modules of simplicial complexes pdf
    (with M. Aprodu, C. Raicu, A. Sammartano and A. Suciu)
    To appear in the Journal of Algebraic Combinatorics

    The Minimal Resolution Property for points on general curves pdf
    (with E. Larson)
    To appear in the Annales Scientifiques de l'Ecole Normale Superieure

    Koszul modules with vanishing resonance in algebraic geometry pdf
    (with M. Aprodu, C. Raicu and J. Weyman)
    To appear in Selecta Mathematica

    Difference varieties and the Green-Lazarsfeld Secant Conjecture pdf
    To appear in the special EPIGA issue dedicated to C. Voisin

    The non-abelian Brill-Noether divisor on $\overline{\mathcal{M}}_{13}$ and the Kodaira dimension of $\overline{\mathcal{R}}_{13}$ pdf
    (with D. Jensen and S. Payne)
    Accompanying Maple file,
    here.
    To appear in Geometry & Topology

    Linear series on general curves with prescribed incidence conditions pdf
    (with C. Lian)
    Journal of the Institute of Mathematics of Jussieu 22 (2023), 2857-2877

    Generalized de Jonquières divisors on generic curves pdf
    Studia Univ. Babes-Bolyai Math. 68 (2023), 13-27 (special volume dedicated to the memory of Cs. Varga)

    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)
    Pure and Applied Mathematics Quarterly (Clemens volume) 18 (2022), 1211-1263

    Topological invariants of groups and Koszul modules pdf
    (with M. Aprodu, S. Papadima, C. Raicu and J. Weyman)
    Duke Mathematical Journal 171 (2022), 2013-2046

    On the Kodaira dimension of Hurwitz spaces pdf
    (with S. Mullane)
    Mathematische Zeitschrift 300 (2022), 3417-3432 (special volume dedicated to O. Debarre)

    Effective divisors on Hurwitz spaces pdf
    Facets of Algebraic Geometry (LMS Lecture Notes Series 472), Vol. 1, 221-240, a volume in honor of Fulton's 80th birthday

    The unirationality of the moduli space of K3 surfaces of genus 22 pdf
    (with A. Verra)
    Mathematische Annalen 380 (2021), 953-973

    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

    The universal difference variety over $\mathcal{M}_g$ pdf
    (with A. Verra) Rendiconti del Circolo Matematico di Palermo 62 (2013), 97-110

    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
    Arie Peterson (PhD 2015), Modular forms on the moduli space of polarized K3 surfaces
    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
    Carlos Maestro Perez (PhD 2021), Aspects of the geometry of Prym varieties and their moduli
    Andrei Bud (PhD 2022), Moduli of Prym curves and strata of differentials - MATH+ Dissertation Award 2024 and Finalist of the Adlershofer Dissertationspreis 2023

    Current PhD students: Li Li (2021-), Federico Moretti (2020-)