Prof. Dr. Gavril Farkas

Profil

• Einstein Visiting Fellowship Rahul Pandharipande – Moduli of Curves, Bundles and K3 Surfaces

  Quelle ↗

  Förderer: Einstein Visiting Fellow Zeitraum: 01/2015 - 12/2019 Projektleitung: Prof. Dr. Gavril Farkas

• EXC 2046/1: Algebraic and Arithmetic Geometry

  Quelle ↗

  Förderer: DFG Exzellenzstrategie Cluster Zeitraum: 05/2023 - 12/2025 Projektleitung: Prof. Dr. Gavril Farkas

• GRK 2965/1: „Von Geometrie zu Zahlen: Moduli, Hodge Theorie, rationale Punkte“

  Quelle ↗

  Förderer: DFG Graduiertenkolleg Zeitraum: 10/2024 - 09/2029 Projektleitung: Prof. Dr. Gavril Farkas

• GRK 2965: Von Geometrie zu Zahlen: Moduli, Hodge Theorie, rationale Punkte

  Quelle ↗

  Förderer: DFG Graduiertenkolleg Zeitraum: 10/2024 - 09/2029 Projektleitung: Prof. Dr. Stefan Schreieder

• IGRK 1800/1: Moduli and Automorphic Forms: Arithmetic and Geometric Aspects

  Quelle ↗
  409-01-A · Algorithmik und Komplexität

  Förderer: DFG Graduiertenkolleg Zeitraum: 07/2012 - 12/2016 Projektleitung: Prof. Dr. phil. Jürg Kramer

• Kolleg Mathematik Physik Berlin

  Quelle ↗

  Zeitraum: 01/2016 - 12/2020 Projektleitung: Prof. Dirk Kreimer, Prof. Dr. Gavril Farkas, Prof. Dr. Jan Plefka

• SFB 647/1-2: Strings, D-Branes und Mannigfaltigkeiten von spezieller Holonomie (TP A01)

  Quelle ↗

  Förderer: DFG Sonderforschungsbereich Zeitraum: 01/2005 - 06/2009 Projektleitung: Prof. Dr. Gavril Farkas

• SFB 647/2-3: Enumerative Geometry of Moduli Spaces (TP A 09/C 03)

  Quelle ↗

  Förderer: DFG Sonderforschungsbereich Zeitraum: 01/2009 - 12/2016 Projektleitung: Prof. Dr. Gavril Farkas

• SPP 1489: Syzygies, Hurwitz Spaces and Ulrich Sheaves

  Quelle ↗

  Förderer: DFG Schwerpunktprogramm Zeitraum: 08/2013 - 09/2017 Projektleitung: Prof. Dr. Gavril Farkas

• SPP 1748: Approximation und Rekonstruktion von Spannungen in der Momentankonfiguration für hyperelastische Materialmodelle

  Quelle ↗

  Förderer: DFG Sachbeihilfe Zeitraum: 09/2019 - 08/2022 Projektleitung: Prof. Dr. Gavril Farkas

• Syzygien und Moduli

  Quelle ↗

  Förderer: DFG Sachbeihilfe Zeitraum: 10/2018 - 08/2023 Projektleitung: Prof. Dr. Gavril Farkas

• Syzygies, Moduli and Topological Invariants of Groups (SYZYGY)

  Quelle ↗

  Förderer: Horizon 2020: ERC Advanced Grant Zeitraum: 03/2020 - 12/2026 Projektleitung: Prof. Dr. Gavril Farkas

• The Birational Geometry of Moduli Space of Curves via Macaulay2

  Quelle ↗

  Förderer: DFG Sachbeihilfe Zeitraum: 07/2010 - 06/2013 Projektleitung: Prof. Dr. Gavril Farkas

• Europäische Kommission

  PT

  87 Treffer56.0%
  • Generalized Quatum Batalin-Vilkovisky Formalism and Graphical Calculus
    P

    56.0%

• Gesellschaft für angewandte Mathematik und Mechanik

  PT

  83 Treffer55.5%
  • Workshop Reliable Methods and Mathematical Modeling
    P

    55.5%

• Magwel NV

  PT

  30 Treffer54.6%
  • EU: Simulation in Multiscale Physical and Biological Systems (STIMULATE)
    T

    54.6%

• NVIDIA GmbH

  PT

  29 Treffer54.6%
  • EU: Simulation in Multiscale Physical and Biological Systems (STIMULATE)
    T

    54.6%

• International Business Machines Corporation Research GmbH

  PT

  36 Treffer54.6%
  • EU: Simulation in Multiscale Physical and Biological Systems (STIMULATE)
    T

    54.6%
  • EU: Bottom-Up Generation of atomicalLy Precise syntheTIc 2D MATerials for High Performance in Energy and Electronic Applications – A Multi-Site Innovative Training Action (ULTIMATE)
    T

    44.6%

• Bernstein Center for Computational Neuroscience Berlin

  PT

  62 Treffer54.2%
  • DFG-Sachbeihilfe: Aufmerksamkeit und sensorische Integration im aktiven Sehen von bewegten Objekten
    T

    54.2%
  • SFB 1315/2: Mechanismen und Störungen der Gedächtniskonsolidierung: Von Synapsen zur Systemebene
    P

    49.3%

• Science & Startups Berlin

  PT

  14 Treffer53.9%
  • Zuwendung im Rahmen des Programms „exist – Existenzgründungen aus der Wissenschaft“ aus dem Bundeshaushalt, Einzelplan 09, Kapitel 02, Titel 68607, Haushaltsjahr 2026, sowie aus Mitteln des Europäischen Strukturfonds (hier Euro-päischer Sozialfonds Plus – ESF Plus) Förderperiode 2021-2027 – Kofinanzierung für das Vorhaben: „exist Women“
    T

    53.9%

• InnovationLab GmbH

  PT

  23 Treffer53.7%
  • Interfaces in opto-electronic thin film multilayer devices
    P

    53.7%

• RISC Software GmbH

  PT

  58 Treffer53.7%
  • EU: Scattering Amplitudes: From Geometry to EXperiment (SAGEX)
    P

    53.7%

• SoftBank Robotics

  P

  32 Treffer51.9%
  • Embodied Audition for RobotS
    P

    51.9%

• Koszul divisors on moduli spaces of curves

  2009

  American Journal of Mathematics · 119 Zitationen · DOI

  Given a moduli space, how can one construct the ``best'' (in the sense of higher dimensional algebraic geometry) effective divisor on it? We show that, at least in the case of the moduli space of curves, the answer is provided by the Koszul divisor defined in terms of the syzygies of the parameterized objects. In this paper, we find a formula for the slopes of all Koszul divisors on $\overline{{\cal M}}_g$. In particular, we obtain the first infinite series of counterexamples to the Harris-Morrison Slope Conjecture and we prove the Maximal Rank Conjecture in the case when the Brill-Noether number of the corresponding linear series equals~$0$. We also find shorter proofs for the formulas of the class of the Brill-Noether and Gieseker-Petri divisors. Finally, we improve most of Logan's results on the Kodaira dimension of the moduli spaces $\overline{{\cal M}}_{g, n}$ of pointed stable curves.

• The Kodaira dimension of the moduli space of Prym varieties

  2010

  Journal of the European Mathematical Society · 79 Zitationen · DOI

  We study the enumerative geometry of the moduli space \mathcal R_g of Prym varieties of dimension g–1 . Our main result is that the compactication of \mathcal R_g is of general type as soon as g > 13 and g is different from 15 . We achieve this by computing the class of two types of cycles on \mathcal R_g : one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym–Green conjecture on syzygies of Prym-canonical curves. We also perform a detailed study of the singularities of the Prym moduli space, and show that for g ≥4 , pluricanonical forms extend to any desingularization of the moduli space.

• Effective divisors on \overline{ℳ}_{ℊ}, curves on 𝒦3 surfaces, and the slope conjecture

  2004

  Journal of Algebraic Geometry · 75 Zitationen · DOI

  We compute the class of the compactification of the divisor of curves sitting on a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Baseline 3"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">K3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surface and show that it violates the Harris-Morrison Slope Conjecture. We carry this out using the fact that this divisor has four distinct incarnations as a geometric subvariety of the moduli space of curves. We also give a counterexample to a hypothesis raised by Harris and Morrison that the Brill-Noether divisors are essentially the only effective divisors on the moduli space of curves having minimal slope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="6 plus 12 slash left-parenthesis g plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>6</mml:mn> <mml:mo>+</mml:mo> <mml:mn>12</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">6+12/(g+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

• The moduli space of twisted canonical divisors

  2018

  Repository for Publications and Research Data (ETH Zurich) · 70 Zitationen · DOI

  ISSN:1474-7480

• The geometry of the moduli space of curves of genus 23

  2000

  Mathematische Annalen · 62 Zitationen · DOI

• Effective divisors on \barM_g, curves on K3 surfaces and the Slope Conjecture

  2003

  arXiv (Cornell University) · 54 Zitationen

• The Mori cones of moduli spaces of pointed curves of small genus

  2002

  Transactions of the American Mathematical Society · 53 Zitationen · DOI

  We compute the Mori cones of the moduli spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M overbar Subscript g comma n"> <mml:semantics> <mml:msub> <mml:mover> <mml:mi>M</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\overline M_{g,n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> pointed stable curves of genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are relatively small. For instance we show that for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g greater-than 14"> <mml:semantics> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>14</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">g&gt;14</mml:annotation> </mml:semantics> </mml:math> </inline-formula> every curve in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M overbar Subscript g"> <mml:semantics> <mml:msub> <mml:mover> <mml:mi>M</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mi>g</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\overline M_g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is equivalent to an effective combination of the components of the locus of curves with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 g minus 4"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>g</mml:mi> <mml:mo> − </mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">3g-4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> nodes. We completely describe the cone of nef divisors for the space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M overbar Subscript 0 comma 6"> <mml:semantics> <mml:msub> <mml:mover> <mml:mi>M</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\overline M_{0,6}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , thus verifying Fulton’s conjecture for this space. Using this description we obtain a classification of all the fibrations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M overbar Subscript 0 comma 6"> <mml:semantics> <mml:msub> <mml:mover> <mml:mi>M</mml:mi> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\overline M_{0,6}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

• Minimal resolutions, Chow forms and Ulrich bundles on <i>K</i>3 surfaces

  2015

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 49 Zitationen · DOI

  Abstract The Minimal Resolution Conjecture (MRC) for points on a projective variety X predicts that the Betti numbers of general sets of points in X are as small as the geometry (Hilbert function) of X allows. To a large extent, we settle this conjecture for a curve C with general moduli. We then proceed to find a full solution to the Ideal Generation Conjecture for curves with general moduli. In a different direction, we prove that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo>⁢</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> K3 surfaces admit Ulrich bundles of every rank. We apply this to describe a pfaffian equation for the Chow form of a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo>⁢</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> K3 surface.

• The geometry of the moduli space of odd spin curves

  2014

  Annals of Mathematics · 49 Zitationen · DOI

  The spin moduli space Sg is the parameter space of theta characteristics (spin structures) on stable curves of genus g. It has two connected components, S -

• The birational type of the moduli space of even spin curves

  2009

  Advances in Mathematics · 46 Zitationen · DOI

• Gaussian maps, Gieseker-Petri loci and large theta-characteristics

  2005

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 38 Zitationen · DOI

  For an integer g ≥ 1 we consider the moduli space Sg of smooth spin curves parametrizing pairs (C,L), where C is a smooth curve of genus g and L is a thetacharacteristic, that is, a line bundle on C such that L2 ∼ = KC. It has been known classically that the natural map π: Sg → Mg is finite of degree 22g and that Sg is a disjoint union

• Moduli of theta-characteristics via Nikulin surfaces

  2011

  Mathematische Annalen · 37 Zitationen · DOI

• The classification of universal Jacobians over the moduli spaces of curves

  2013

  Iris (Roma Tre University) · 35 Zitationen · DOI

  We complete the classification by Kodaira dimension of the moduli space of degree g line bundles over curves of genus g , for all genera.

• Syzygies of curves and the effective cone of Mg

  2006

  Duke Mathematical Journal · 35 Zitationen · DOI

  We describe a systematic way of constructing effective divisors on the moduli space of stable curves having exceptionally small slope. We show that every codimension 1 locus in M̲g consisting of curves failing to satisfy a Green-Lazarsfeld syzygy-type condition provides a counterexample to the Harris-Morrison slope conjecture. We also introduce a new geometric stratification of the moduli space of curves somewhat similar to the classical stratification given by gonality but where the analogues of hyperelliptic curves are the sections of K3 surfaces

• Divisors on and the minimal resolution conjecture for points on canonical curves

  2003

  Annales Scientifiques de l École Normale Supérieure · 35 Zitationen · DOI

  We use geometrically defined divisors on moduli spaces of pointed curves to compute the graded Betti numbers of general sets of points on any nonhyperelliptic canonically embedded curve. This gives a positive answer to the Minimal Resolution Conjecture in the case of canonical curves. But we prove that the conjecture fails on curves of large degree. These results are related to the existence of theta divisors associated to certain stable vector bundles. Nous utilisons des diviseurs définis pour des conditions géométriques sur des espaces de modules de courbes stables à points marqués pour calculer les nombres de Betti des ensembles généraux de points sur une courbe non hyperelliptique arbitraire, canoniquement plongée. Cela donne une réponse affirmative à la conjecture de résolution minimale dans le cas des courbes canoniques. Par ailleurs, nous prouvons que la conjecture est fausse pour les courbes de grand degré. Ces résultats sont liés à l'existence des diviseurs thêta associés à certains fibrés vectoriels stables.

• The generic Green–Lazarsfeld Secant Conjecture

  2015

  Inventiones mathematicae · 27 Zitationen · DOI

• Syzygies of torsion bundles and the geometry of the level ℓ modular variety over $\overline{\mathcal{M}}_{g}$

  2012

  Inventiones mathematicae · 27 Zitationen · DOI

• The Maximal Rank Conjecture and Rank Two Brill-Noether Theory

  2011

  Pure and Applied Mathematics Quarterly · 26 Zitationen · DOI

  We describe applications of Koszul cohomology to the Brill-Noether theory of rank 2 vector bundles. Among other things, we show that in every genus g > 10, there exist curves invalidating Mercat's Conjecture for rank 2 bundles. On the other hand, we prove that Mercat's Conjecture holds for general curves of bounded genus, and its failure locus is a Koszul divisor in the moduli space of curves. We also formulate a conjecture concerning the minimality of Betti diagrams of suitably general curves, and point out its consequences to rank 2 Brill-Noether theory.

• The global geometry of the moduli space of curves

  2009

  Proceedings of symposia in pure mathematics · 26 Zitationen · DOI

  We survey the progress made in the last decade in understanding the birational geometry of the moduli space of stable curves. Topics that are being discusses include the cones of ample and effective divisors, Kodaira dimension and minimal models of Mg.

• Higher ramification and varieties of secant divisors on the generic curve

  2008

  Journal of the London Mathematical Society · 25 Zitationen · DOI

  For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry, and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. The aim of this paper is to clarify this problem in the case of the generic curve C of given genus. We determine precisely under which conditions the cycle of e-secant k-planes is non-empty, and we compute its dimension. We also precisely determine the dimension of the variety of linear series on C carrying e-secant k-planes.

• Rational maps between moduli spaces of curves and Gieseker-Petri divisors

  2009

  Journal of Algebraic Geometry · 24 Zitationen · DOI

  We study contractions of the moduli space of stable curves beyond the minimal model of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M overbar Subscript g prime"> <mml:semantics> <mml:msub> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>g</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\overline {\mathcal {M}}_{g’}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by giving a complete enumerative description of the rational map between two moduli spaces of curves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M overbar Subscript g Baseline right dasheD arrow script upper M overbar Subscript g prime"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mi>g</mml:mi> </mml:msub> <mml:mo stretchy="false"> ⇢ </mml:mo> <mml:msub> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>g</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\overline {\mathcal {M}}_g \dashrightarrow \overline {\mathcal {M}}_{g’}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which associates to a curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> its Brill–Noether locus of special divisors in the case this locus is a curve. As an application we construct many examples of moving effective divisors on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M overbar Subscript g"> <mml:semantics> <mml:msub> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mo accent="false"> ¯ </mml:mo> </mml:mover> <mml:mi>g</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\overline {\mathcal {M}}_g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of small slope, which in turn can be used to show that various moduli space of curves with level structure are of general type. For low <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g prime"> <mml:semantics> <mml:msup> <mml:mi>g</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">g’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> our calculation can be used to study the intersection theory of the moduli space of Prym varieties of dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="5"> <mml:semantics> <mml:mn>5</mml:mn> <mml:annotation encoding="application/x-tex">5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

• Koszul cohomology and applications to moduli

  2008

  arXiv (Cornell University) · 23 Zitationen · DOI

  We discuss recent progress on syzygies of curves, including proofs of Green's and Gonality Conjectures as well as applications of Koszul cycles to the study of the birational geometry of various moduli spaces of curves. We prove a number of new results, including a complete solution to Green's Conjecture for arbitrary hexagonal curves. Finally, we propose several new conjectures on syzygies, including a Prym-Green conjecture for l-roots of trivial bundles as well as a strong Maximal Rank Conjecture for generic curves. To appear in the Proceedings of Clay Mathematical Institute.

• Prym varieties and moduli of polarized Nikulin surfaces

  2015

  Advances in Mathematics · 21 Zitationen · DOI

• Brill-Noether loci and the gonality stratification of {M_g}

  2001

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 21 Zitationen · DOI

  Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 Rome / CNR - Consiglio Nazionale delle Richerche

• Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces

  2012

  arXiv (Cornell University) · 20 Zitationen · DOI

  The Minimal Resolution Conjecture (MRC) for points on a projective variety X predicts that the Betti numbers of general sets of points in X are as small as the geometry (Hilbert function) of X allows. To a large extent, we settle this conjecture for a curve C with general moduli. We show that, independently of the genus, MRC holds for a general linear system of degree d and dimension r on C if and only if d&gt;2r-1. We then proceed to find a full solution to the Ideal Generation Conjecture for curves with general moduli. In a different direction, we prove that K3 surfaces admit Ulrich bundles of every rank. We apply this to describe a pfaffian equation for the Chow form of a K3 surface.

• Gottfried Wilhelm Leibniz Universität Hannover

  GRK 2965: Von Geometrie zu Zahlen: Moduli, Hodge Theorie, rationale Punkte

  university

• Universität Duisburg-Essen

  SPP 1748: Approximation und Rekonstruktion von Spannungen in der Momentankonfiguration für hyperelastische Materialmodelle

  university

Name

Prof. Dr. Gavril Farkas

Titel

Prof. Dr.

Fakultät

Mathematisch-Naturwissenschaftliche Fakultät

Institut

Institut für Mathematik

Arbeitsgruppe

Algebraische Geometrie I

Telefon

+49 30 2093-45442

HU-FIS-Profil

Quelle ↗

Zuletzt gescrapt

26.4.2026, 01:04:27