We study the surjectivity of suitable weighted Gaussian maps which provide a natural generalization of the standard Gaussian maps and encode the local geometry of the locus of curves endowed with a higher root of the canonical bundle having associated linear system of dimension at least r + 1. In particular, we get a bound on the dimension of its Zariski tangent space, which turns out to be sharp in the special case r = 0. Finally, we describe this locus in the case of complete intersection curves.
A theorem of Jannsen asserts that if a smooth projective variety has injective cycle class maps, it has surjective cycle class maps. The object of this note is to present a version of Jannsen's theorem for singular quasi--projective varieties.
Making suitable generalizations of known results we prove some general facts about Gaussian maps. The above are then used, in the second part of the article, to give a set of conditions that insure the surjectivity of Gaussian maps for curves on Enriques surfaces. To do this we also solve a problem of independent interest: a tetragonal curve of genus g > 6 lying on an Enriques surface and general in its linear system, cannot be, in its canonical embedding, a quadric section of a surface of degree g - 1 in P^{g - 1}.
This paper explores the geometric meaning of the failure of certain kinds of Wahl maps to surject on a general curve. Sufficient conditions for surjectivity are given. An approach used by Voisin to study canonical Wahl maps is applied in this direction.
For a positive integer $k$, we extend the surjectivity results from special linear groups (Type $A_k$) and symplectic linear groups (Type $C_k$) onto product of generalized projective spaces by associating the rows or columns, to certain congruence ideal subgroups of special linear groups and symplectic linear groups.
We prove in this article the surjectivity of three maps. We prove in Theorem $1.6$ the surjectivity of the Chinese remainder reduction map associated to the projective space of an ideal with a given factorization into ideals whose radicals are pairwise distinct maximal ideals. In Theorem $1.7$ we prove the surjectivity of the reduction map of the strong approximation type for a ring quotiented by an ideal which satisfies unital set condition. In Theorem $1.8$ we prove for Dedekind type domains which include Dedekind domains, for $k\geq 2$, the map from $k$-dimensional special linear group to t…
We show that an element w of a free group F on n generators defines a surjective word map of PSL(2,C)^n onto PSL(2,C) unless w belongs to the second derived subgroup of F. We also describe certain words maps that are surjective on SL(2,C) x SL(2,C). Here C is the field of complex numbers.
In this paper, we establish two results concerning algebraic $(\mathbb{C},+)$-actions on $\mathbb{C}^n$. First let $ϕ$ be an algebraic $(\mathbb{C},+)$-action on $\mathbb{C}^3$. By a result of Miyanishi, its ring of invariants is isomorphic to $\mathbb{C}[t_1,t_2]$. If $f_1,f_2$ generate this ring, the quotient map of $ϕ$ is the map $F:\mathbb{C}^3\to \mathbb{C}^2$, $x\mapsto (f_1(x),f_2(x))$. By using some topological arguments, we prove that $F$ is always surjective. Second, we are interested in dominant polynomial maps $F:\mathbb{C}^n\to \mathbb{C}^{n-1}$ whose connected components of their…
In this article we introduce generalized projective spaces (Definitions $[2.1, 2.5]$) and prove three main theorems in two different contexts. In the first context we prove, in main Theorem $A$, the surjectivity of the Chinese remainder reduction map associated to the generalized projective space of an ideal with a given factorization into mutually co-maximal ideals each of which is contained in finitely many maximal ideals, using the key concept of choice multiplier hypothesis (Definition $4.11$) which is satisfied. In the second context of surjectivity of the map from $k$-dimensional special…
In this article, for generalized projective spaces with any weights, we prove four main theorems in three different contexts where the Unital Set Condition USC (Definition $2.8$) on ideals is further examined. In the first context we prove, in the first main Theorem $A$, the surjectivity of the Chinese remainder reduction map associated to the generalized projective space of an ideal $\mathcal{I}=\underset{i=1}{\overset{k}{\prod}}\mathcal{I}_k$ with a given factorization into mutually co-maximal ideals $\mathcal{I}_j,1\leq j\leq k$ where $\mathcal{I}$ satisfies the USC, using the key concept o…
Let $S_{g}$ be a general prime K3 surface in $P^g$ of genus $g \geq 3$ or a general double cover of $P^2$ ramified along a sextic curve for $g = 2$ and $S = S_{i,g}$ its {\it i}-th Veronese embedding. In this article we compute the corank of the Gaussian map $Φ:\bigwedge^2 H^0(S,O_S(1)) \to H^0(S,Ω_S^1(2))$ for $i \geq 2, g \geq 2$ and $i=1, g \geq 17$. The main idea is to reduce the surjectivity of $Φ$ to an application of the Kawamata-Viehweg vanishing theorem on the blow-up of $S \times S$ along,the diagonal. This is seen to apply once the hyperplane divisor of the K3 surface $S$ can be dec…
We study the intersection theory on the moduli spaces of maps of $n$-pointed curves $f:(C,s_1,... s_n)\to V$ which are stable with respect to a weight data $(a_1,..., a_n)$, $0\le a_i\le 1$. After describing the structure of these moduli spaces, we prove a formula describing the way each descendant changes under a wall crossing. As a corollary, we compute the weighted descendants in terms of the usual ones, i.e. for the weight data $(1,...,1)$, and vice versa.
We consider a class of Calabi-Yau compactifications which are constructed as a complete intersection in weighted projective space. For manifolds with one Kähler modulus we construct the mirror manifolds and calculate the instanton sum.
We classify birational maps into elliptic fibrations of a general quasismooth hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ that has terminal singularities.
In this thesis we study singular curves on K3 surfaces. Let $\mathcal{B}_g$ denote the stack of polarised K3 surfaces of genus $g$ and set $p(g,k)=k^2(g-1)+1$. There is a stack $ \mathcal{T}^n_{g,k} \to \mathcal{B}_g$ with fibre over the polarised surface $(X,L)$ parametrising all unramified morphisms $f: C \to X$, birational onto their image, with $C$ an integral smooth curve of genus $ p(g,k)-n$ and $f_*C \sim kL$. One can think of $ \mathcal{T}^n_{g,k}$ as parametrising all singular curves on K3 surfaces such that the normalisation map is unramified (or equivalently such that the curve has …
Let $(X,0)\subset (\mathbb{C}^n,0)$ be an irreducible weighted homogeneous singularity curve and let $f:(X,0)\to(\mathbb{C}^2,0)$ be a map germ finite, one-to-one and weighted homogeneous with the same weights of $(X,0)$. We show that $\mathscr{A}_e$-$codim(X,f)=μ_I(f)$, where $\mathscr{A}_e$-$codim(X,f)$ is the $\mathscr{A}_e$-codimension, i.e., the minimum number of parameters in a versal deformation and $μ_I(f)$ is the image Milnor number, i.e., the number of vanishing cycles in the image of a stabilisation of $f$.
We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for projective dimensions of hyperplane arrangements. They are generalizations of the free arrangement cases, that can be regarded as the special case of our result when the projective dimension is zero. The keys to prove them are several new methods to determine the surjectivity of the Euler and the Ziegler restriction maps, that is combinatorial when the projective…
This is a note on my mini-course in the International Workshop on Real and Complex Singularities held at ICMC-USP (Sao Carlos, Brazil) in July 2012. Here we introduce a new branch of the Thom polynomial theory for singularities of holomorphic maps, in which we replace counting singular points by computing weighted Euler characteristics. The main purpose is to apply this theory to the study on the vanishing topology of weighted homogeneous map-germs of finite A-codimension without any corank condition.
