Marian Aprodu

Bio: Marian Aprodu is an academic researcher from University of Bucharest. The author has contributed to research in topics: Conjecture & Vector bundle. The author has an hindex of 17, co-authored 72 publications receiving 880 citations. Previous affiliations of Marian Aprodu include Max Planck Society & University of Bayreuth.

Koszul Cohomology and Algebraic Geometry

Abstract: The systematic use of Koszul cohomology computations in algebraic geometry can be traced back to the foundational work of Mark Green in the 1980s. Green connected classical results concerning the ideal of a projective variety with vanishing theorems for Koszul cohomology. Green and Lazarsfeld also stated two conjectures that relate the Koszul cohomology of algebraic curves with the existence of special divisors on the curve. These conjectures became an important guideline for future research. In the intervening years, there has been a growing interaction between Koszul cohomology and algebraic geometry. Green and Voisin applied Koszul cohomology to a number of Hodge-theoretic problems, with remarkable success. More recently, Voisin achieved a breakthrough by proving Green's conjecture for general curves; soon afterwards, the Green-Lazarsfeld conjecture for general curves was proved as well. This book is primarily concerned with applications of Koszul cohomology to algebraic geometry, with an emphasis on syzygies of complex projective curves. The authors' main goal is to present Voisin's proof of the generic Green conjecture, and subsequent refinements. They discuss the geometric aspects of the theory and a number of concrete applications of Koszul cohomology to problems in algebraic geometry, including applications to Hodge theory and to the geometry of the moduli space of curves.

Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces

Abstract: The Minimal Resolution Conjecture (MRC) for points on a projective varietyX ⊂ P predicts that the minimal graded free resolution of a general set Γ ⊂ X of points is as simple as the geometry of X allows. Originally, the most studied case has been that when X = P, see [EPSW]. The general form of the MRC for subvarieties X ⊂ P was formulated in [Mus] and [FMP]. The Betti diagram of a large enough set Γ ⊂ X consisting of γ general points is obtained from the Betti diagram of X , by adding two rows, indexed by u − 1 and u, where u is an integer depending on γ. All differences bi+1,u−1(Γ)− bi,u(Γ) are known and depend on the Hilbert polynomial PX and i, u and γ, see [FMP]. The Minimal Resolution Conjecture for γ general points on X predicts that

Green's Conjecture for curves on arbitrary K3 surfaces

Abstract: Green's Conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin's results on syzygies of K3 sections, to the case of K3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz-Ramanan, provides a complete solution to Green's Conjecture for smooth curves on arbitrary K3 surfaces.

Green’s conjecture for curves on arbitrary K3 surfaces

Abstract: Green’s conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin’s results on syzygies of K3 sections, to the case of K3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz–Ramanan, provides a complete solution to Green’s conjecture for smooth curves on arbitrary K3 surfaces.

Remarks on syzygies of d-gonal curves

Abstract: We apply a degenerate version of a result due to Hirschowitz, Ramanan and Voisin to verify Green and Green-Lazarsfeld conjectures over explicit open sets inside each $d$-gonal stratum of curves $X$ with $d<[g_X/2]+2$. By the same method, we verify the Green-Lazarsfeld conjecture for any curve of odd genus and maximal gonality. The proof invokes Voisin's solution to the generic Green conjecture as a key argument.

Green's generic syzygy conjecture for curves of even genus lying on a K3 surface

Abstract: for X a variety and L a line bundle on X. Denoting by Kp,q(X,L) the cohomology at the middle of the sequence above, one sees immediately that the surjectivity of the map μ0 is equivalent to K0,2(C, KC) = 0, and that if this is the case, the ideal I is generated by quadrics if and only if K1,2(C, KC) = 0. On the other hand, C being non hyperelliptic is equivalent to the fact that the Clifford index Cliff C is strictly positive, where

green's canonical syzygy conjecture for generic curves of odd genus

Abstract: Kl,1(C, KC) = 0, ∀l ≥ p ⇔ Cliff(C) > g − p− 2. The direction ⇒ is proved by Green and Lazarsfeld in the appendix to [4]. The case p = g − 2 of the conjecture is equivalent to Noether’s theorem, and the case p = g − 3 to Petri’s theorem (see [6]). The case p = g − 4 has been proved in any genus by Schreyer [10] and by the author [13] for g > 10. More recently, the conjecture has been studied in [11], [12], for generic curves of fixed gonality. Teixidor proves the following

Differential geometry of singular spaces and reduction of symmetry

Abstract: Preface 1. Introduction Part I. Differential Geometry of Singular Spaces: 2. Differential structures 3. Derivations 4. Stratified spaces 5. Differential forms Part II. Reduction of Symmetries: 6. Symplectic reduction 7. Commutation of quantization and reduction 8. Further examples of reduction References Index.