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In this paper we consider the Brill-Noether locus $W_{underline d}(C)$ of line bundles of multidegree $underline d$ of total degree $g-1$ having a nonzero section on a nodal reducible curve $C$ of genus $ggeq2$. We give an explicit description of the irreducible components of $W_{underline d}(C)$ for a semistable multidegre $underline d$. As a consequence we show that, if two semistable multidegrees of total degre $g-1$ on a curve with no rational components differ by a twister, then the respective Brill-Noether loci have isomorphic components.
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We compute the integral cohomology groups of the smooth Brill-Noether varieties $G^r_d(C)$, parametrizing linear series of degree $d$ and dimension exactly $r$ on a general curve $C$. As an application, we determine the whole intersection cohomology of the singular Brill-Noether loci $W^r_d(C)$, parametrizing complete linear series on $C$ of degree $d$ and dimension at least $r$.
We slightly extend a previous result concerning the injectivity of a map of moduli spaces and we use this result to construct curves whose Brill-Noether loci have unexpected dimension.
Let $C$ be a curve of genus $g$. A fundamental problem in the theory of algebraic curves is to understand maps $C to mathbb{P}^r$ of specified degree $d$. When $C$ is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. Despite much study over the past three decades, a similarly complete picture has proved elusive for curves of fixed gonality. Here we complete such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting. As a corollary, we prove a conjecture of Eisenbud and Schreyer regarding versal deformation spaces of vector bundles on $mathbb{P}^1$.
Let $Z$ be a closed subscheme of a smooth complex projective variety $Ysubseteq Ps^N$, with $dim,Y=2r+1geq 3$. We describe the intermediate Neron-Severi group (i.e. the image of the cycle map $A_r(X)to H_{2r}(X;mathbb{Z})$) of a general smooth hypersurface $Xsubset Y$ of sufficiently large degree containing $Z$.
Let $K$ be an algebraically closed field of characteristic different from $2$, $g$ a positive integer, $f(x)in K[x]$ a degree $2g+1$ monic polynomial without repeated roots, $C_f: y^2=f(x)$ the corresponding genus g hyperelliptic curve over $K$, and $J$ the jacobian of $C_f$. We identify $C_f$ with the image of its canonical embedding into $J$ (the infinite point of $C_f$ goes to the zero of group law on $J$). It is known (arXiv:1809.03061 [math.AG]) that if $g>1$ then $C_f(K)$ does not contain torsion points, whose order lies between $3$ and $2g$. In this paper we study torsion points of order $2g+1$ on $C_f(K)$. Despite the striking difference between the cases of $g=1$ and $g> 1$, some of our results may be viewed as a generalization of well-known results about points of order $3$ on elliptic curves. E.g., if $p=2g+1$ is a prime that coincides with $char(K)$, then every odd degree genus $g$ hyperelliptic curve contains, at most, two points of order $p$. If $g$ is odd and $f(x)$ has real coefficients, then there are, at most, two real points of order $2g+1$ on $C_f$. If $f(x)$ has rational coefficients and $g<52$, then there are, at most, two rational points of order $2g+1$ on $C_f$. (However, there are exist genus $52$ hyperelliptic curves over the field of rational numbers that have, at least, four rational points of order 105.)
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