Conjecture For Curves Research Articles

Green’s Conjecture for curves on rational surfaces with an anticanonical pencil

Green’s Conjecture is proved for smooth curves $$C$$ lying on a rational surface $$S$$ with an anticanonical pencil, under some mild hypotheses on the line bundle $$L=\mathcal{O }_S(C)$$ . Constancy of Clifford dimension, Clifford index and gonality of curves in the linear system $$\vert L\vert $$ is also obtained.

Green’s conjecture for curves on arbitrary K3 surfaces

AbstractGreen’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.

Critical L-values of Level p Newforms (mod p)

Suppose that p≥ 5 is prime, that ℱ(z) ∈ S 2k (Γ 0 (p)) is a newform, that v is a prime above p in the field generated by the coefficients of ℱ, and that D is a fundamental discriminant. We prove non-vanishing theorems modulo v for the twisted central critical values L(ℱ⊗ χ D ,k). For example, we show that if k is odd and not too large compared to p, then infinitely many of these twisted L-values are non-zero (mod v). We give applications for elliptic curves. For example, we prove that if E/ℚ is an elliptic curve of conductor p, where p is a sufficiently large prime, there are infinitely many twists D with Ш(E D /ℚ)[p] = 0, assuming the Birch and Swinnerton-Dyer conjecture for curves of rank zero as well as a weak form of Hall’s conjecture. The results depend on a careful study of the coefficients of half-integral weight newforms of level 4p, which is of independent interest.

Chisini's conjecture for curves with singularities of type xn=ym

Chisini's conjecture for curves with singularities of type xn=ym

Nakai’s conjecture for varieties smoothed by normalization

The notion of D-simplicity is used to give a short proof that varieties whose normalization is smooth satisfy Ishibashi’s extension of Nakai’s conjecture to arbitrary characteristic. This gives a new proof of Nakai’s conjecture for curves and Stanley-Reisner rings.

The Artinian Berger Conjecture

We propose an Artinian version of Berger's Conjecture for curves, concerning the module of Kahler differentials of an algebra. Our version implies Berger's Conjecture in characteristic 0. We establish our Artinian Berger Conjecture in a number of cases, and prove that Berger's Conjecture holds for curve singularities whose conductor ideal contains the cube of a maximal ideal.