The aim of this paper is to show that the quasihomogeneity of a quasihomogeneous germ with an isolated singularity uniquely extends to the base of its analytic miniversal deformation.

We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in ${}_q\left[T\right]$ of degree d for which s consecutive coefficients $a=(a_{d-1},...,a_{d-s})$ are fixed. Our estimate asserts that $(d,s,a)={\mu }_{d}q+\left(q^{1/2}\right)$, where ${\mu }_d:={\sum }_{r=1}^d\left({(-1)}^{r-1}\right)/(r!)$. We also prove that $₂(d,s,a)=\mu {²}_{d}q²+\left(q^{3/2}\right)$, where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of ${}_q\left[T\right]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $₂(d,0)=\mu {²}_{d}q²+\left(q\right)$, where ₂(d,0) denotes the average second moment for all monic polynomials...

Let $Y$ be an integral projective curve with $g:=p_a\left(Y\right)\ge 2$. For all positive integers $d$, $r$ let $W_d^r\left(Y\right)\left({}^{*A*}\right)$ be the set of all $L\in {\text{Pic}}^d\left(Y\right)$ with $h^0\left(Y,L\right)\ge r+1$ and $L$ spanned. Here we prove that if $d\le g-2$, then $dim\left(W_d^r\left(Y\right)\left({}^{*A*}\right)\right)\le d-3r$ except in a few cases (essentially if $Y$ is a double covering).

Sia $X\subset \mathbb{P}\left(H^0{\left(X,L\right)}^{*}\right)$ una curva proeittiva e lissa, generali nel senso di Brill-Noether, indichiamo con ${\mathcal{R}}_4\left(X\right)$ l'insieme algebrico di quadrici di rango $4$ contenendo a $X$. In questo lavoro noi descriviamo birazionalmente i componenti irriducibile di ${\mathcal{R}}_4\left(X\right)$.

Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<(nn+x). Here we prove that the order x Veronese embedding ofP n is not weakly (k−1)-defective, i.e. for a general S⊃P n such that #(S) = k+1 the projective space | I 2S (x)| of all degree t hypersurfaces ofP n singular at each point of S has dimension (n/n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I 2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S.

Let $Y$ be a normal crossing divisor in the smooth complex projective algebraic variety $X$ and let $U$ be a tubular neighbourhood of $Y$ in $X$. Using geometrical properties of different intersections of the irreducible components of $Y$, and of the embedding $Y\subset X$, we provide the “normal forms” of a set of geometrical cycles which generate $H_{*}(A,B)$, where $(A,B)$ is one of the following pairs $(Y,\varnothing )$, $(X,Y)$, $(X,X-Y)$, $(X-Y,\varnothing )$ and $(\partial U,\varnothing )$. The construction is compatible with the weights in $H_{*}(A,B,\mathbb{Q})$ of Deligne’s mixed Hodge structure. The main technical part...