On the Hilbert function of intersections of a hypersurface with general reducible curves

Let $W\subset \mathbb {P}^n$, $n\ge 3$, be a degree $k$ hypersurface. Consider a "general" reducible, but connected, curve $Y\subset \mathbb {P}^n$, for instance a sufficiently general connected and nodal union of lines with $p_a(Y)=0$, i.e. a tree of lines. We study the Hilbert function of the set $Y\cap W$ with cardinality $k\deg (Y)$ and prove when it is the expected one. We give complete classification of the exceptions for $k=2$ and for $n=k=3$. We apply these results and tools to the case in which $Y$ is a smooth curve with $\mathcal {O}_Y(1)$ non-special.Comment: corrected a big typo in the first two lines of the introduction, no other modificatio

On the gonality sequence of smooth curves: normalizations of singular curves in a quadric surface

Let $C$ be a smooth curve of genus $g$. For each positive integer $r$ the $r$-gonality $d_r(C)$ of $C$ is the minimal integer $t$ such that there is $L\in {Pic}^t(C)$ with $h^0(C,L) =r+1$. In this paper for all $g\ge 40805$ we construct several examples of smooth curves $C$ of genus $g$ with $d_3(C)/3< d_4(C)/4$, i.e. for which a slope inequality fails.Comment: Accepted on Acta Math. Vie

Embeddings of general curves in projective spaces: the range of the quadrics

Let $C \subset \mathbb {P}^r$ a general embedding of prescribed degree of a general smooth curve with prescribed genus. Here we prove that either $h^0(\mathbb {P}^r,\mathcal {I}_C(2)) =0$ or $h^1(\mathbb {P}^r,\mathcal {I}_C(2)) =0$ (a problem called the Maximal Rank Conjecture in the range of quadrics)

Ranks on the boundaries of secant varieties

In many cases (e.g. for many Segre or Segre embeddings of multiprojective spaces) we prove that a hypersurface of the $b$-secant variety of $X\subset \mathbb {P}^r$ has $X$-rank $>b$. We prove it proving that the $X$-rank of a general point of the join of $b-2$ copies of $X$ and the tangential variety of $X$ is $>b$

On the typical rank of real bivariate polynomials

Here we study the typical rank for real bivariate homogeneous polynomials of degree $d\ge 6$ (the case $d\le 5$ being settled by P. Comon and G. Ottaviani). We prove that $d-1$ is a typical rank and that if $d$ is odd, then $(d+3)/2$ is a typical rank

On the stratification by XX-ranks of a linearly normal elliptic curve X⊂PnX\subset \mathbb {P}^n

Let $X\subset \mathbb {P}^n$ be a linearly normal elliptic curve. For any $P\in \mathbb {P}^n$ the $X$-rank of $P$ is the minimal cardinality of a set $S\subset X$ such that $P\in \langle S\rangle$. In this paper we give an almost complete description of the stratification of $\mathbb {P}^n$ given by the $X$-rank and the open $X$-rank.Comment: Added a result on the open ran

Dependent subsets of embedded projective varieties

Let $X\subset \mathbb {P}^r$ be an integral and non-degenerate variety. Set $n:= \dim (X)$. Let $\rho (X)''$ be the maximal integer such that every zero-dimensional scheme $Z\subset X$ smoothable in $X$ is linearly independent. We prove that $X$ is linearly normal if $\rho (X)''\ge \lceil (r+2)/2\rceil$ and that $\rho (X)'' < 2\lceil (r+1)/(n+1)\rceil$, unless either $n=r$ or $X$ is a rational normal curve

Nodal curves and components of the Hilbert scheme of curves in Pr\mathbb {P}^r with the expected number of moduli

We study the existence of components with the expected number of moduli of the Hilbert scheme of integral nodal curves $C \subset \mathbb {P}^r$ with prescribed degree, arithmetic genus and number of singular points

On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank

Let $\sigma_b(X_{m,d}(\mathbb {C}))(\mathbb {R})$, $b(m+1) < \binom{m+d}{m}$, denote the set of all degree $d$ real homogeneous polynomials in $m+1$ variables (i.e. real symmetric tensors of format $(m+1)\times ... \times (m+1)$, $d$ times) which have border rank $b$ over $\mathbb {C}$. It has a partition into manifolds of real dimension $\le b(m+1)-1$ in which the real rank is constant. A typical rank of $\sigma_b(X_{m,d}(\mathbb {C}))(\mathbb {R})$ is a rank associated to an open part of dimension $b(m+1)-1$. Here we classify all typical ranks when $b\le 7$ and $d, m$ are not too small. For a larger sets of $(m,d,b)$ we prove that $b$ and $b+d-2$ are the two first typical ranks. In the case $m=1$ (real bivariate polynomials) we prove that $d$ (the maximal possible a priori value of the real rank) is a typical rank for every $b$.Comment: Acta Mathematica Vietnaminica (to appear

On the Hilbert function of general unions of curves in projective spaces

Let $X=X_1\cup \cdots \cup X_s\subset \mathbb {P}^n$, $n\ge 4$, be a general union of smooth non-special curves with $X_i$ of degree $d_i$ and genus $g_i$ and $d_i\ge \max \{2g_i-1,g_i+n\}$ if $g_i>0$. We prove that $X$ has maximal rank, i.e. for any $t\in \mathbb {N}$ either $h^0(\mathcal{I}_X(t))=0$ or $h^1(\mathcal{I}_X(t))=0$