Related papers: An algebraic proof of Iitaka's Conjecture C_{2,1}

We construct natural relative compactifications for the relative Jacobian over a family $X/S$ of reduced curves. In contrast with all the available compactifications so far, ours admit a universal sheaf, after an etale base change. Our…

Let C be a projective curve either reduced with planar singularities or contained in a smooth algebraic surface. We show that the canonical ring R(C, \omega_C)= \oplus_{k \geq 0} H^0(C, \omega_C^k is generated in degree 1 if C is…

Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification,…

The Casas-Alvero conjecture states that if $f(X)$ is a monic univariate polynomial of degree $d$ over a characteristic $0$ field $\mathbb{K}$ such that $\gcd(f, f_{i})$ is non-trivial for each $i=1, \dots, d-1$, then $f(X)=(X-\alpha)^d$ for…

Let $E/\mathbf{Q}$ be an elliptic curve and $p\geq 3$ be a prime. We prove the $p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes (i.e., such that the residual representation $E[p]$ is…

The Manin conjecture is established for Ch\^atelet surfaces over Q arising as minimal proper smooth models of the surface Y^2+Z^2=f(X) where f is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not…

Motivated by the study of traces on graph $C^*$-algebras, we consider traces (additive, central maps) on Leavitt path algebras, the algebraic counterparts of graph $C^*$-algebras. In particular, we consider traces which vanish on nonzero…

We consider elliptic curves whose coefficients are degree 2 polynomials in a variable t. We prove that for infinitely many values of t the resulting elliptic curve has rank at least 1. All such curves together form an algebraic surface…

We propose a linear version of the weighted bounded negativity conjecture. It considers a smooth projective surface $X$ over an algebraically closed field of characteristic zero and predicts the existence of a common lower bound on…

We show that any commutative rationally ruled surface with a choice of anticanonical curve admits a 1-parameter family of noncommutative deformations parametrized by the Jacobian of the anticanonical curve, and show that many standard facts…

Miyanishi conjecture claims that for any variety over an algebraically closed field of characteristic zero, any endomorphism of such a variety which is injective outside a closed subset of codimension at least $2$ is bijective. We prove…

Let $K=k(C)$ be the function field of a complete nonsingular curve $C$ over an arbitrary field $k$. The main result of this paper states that a morphism $\phi:{\mathbb P}^N_K\to{\mathbb P}^N_K$ is isotrivial if and only if it has potential…

Let G be a complex reductive group and let C be a smooth curve of genus at least one. We prove a converse to a theorem of Atiyah-Bott concerning the stratification of the space of holomorphic G-bundles on C. In case the genus of C is one,…

Let $A$ be a local noetherian ring and $N$ be a locally sheaf on the projective space $P^3_A$ : one proves easily that there exists a family $C$ of (smooth connected) curves contained in $P^3_A$, flat over $A$, and an integer $h$ such that…

Let $S$ be a K3 surface, $C$ a smooth curve on $S$ with $\mathcal{O}_S(C)$ ample, and $A$ a base-point free $g^2_d$ on $C$ of small degree. We use Lazarsfeld--Mukai bundles to prove that $A$ is cut out by the global sections of a rank 1…

We give refined statements and modern proofs of Rosenlicht's results about the canonical model C' of an arbitrary complete integral curve C. Notably, we prove that C and C' are birationally equivalent if and only if C is nonhyperelliptic,…

The purpose of this note is to give a short proof of a theorem of Koll\'ar that the derived direct image of the canonical sheaf splits into a sum of its cohomology sheaves. This is deduced from a stronger decomposition theorem for direct…

Suppose $Y$ is a smooth variety equipped with a top form. We prove a simple theorem giving a sharp lower bound on the geometric genus of a family of subvarieties of $Y$, in terms of the dimension of this family. Two elementary applications…

In this paper we prove that complete families of smooth and projective curves, of genusg>2, in characteristic p>0, with a constant geometric fundamental group, are isotrivial.

We show the geometric syzygy conjecture in positive characteristic. Specifically, if C is a general smooth curve of genus g defined over an algebraically closed field of characteristic p, then all linear syzygy spaces are spanned by…