# Questions tagged [brill-noether]

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Let $C$ be a smooth projective curve over $\mathbb{C}$. Choose two sets of degree and rank $(d_i,r_i)$ for $i=1,2$. Let $U(d_i,r_i)$ denote the moduli space of vector bundles of degree $d_i$ and rank $... • 916 8 votes 3 answers 583 views ### Modern references to Brill-Noether theory on curves? Are there modern good lecture notes/book about Brill-Noether theory of curves. Most interesting theorems here are proved via limit linear series, which I found no lecture notes on (instead there is ... • 247 2 votes 1 answer 206 views ### Curves having only one linear system realizing its gonality$\DeclareMathOperator\gon{gon}$Let$C$be a smooth irreducible projective curve defined over complex numbers. Recall that the gonality of$C$,$\gon(C)$, is defined to be the minimal possible degree ... • 259 2 votes 1 answer 155 views ### What are the possible Clifford functions of a curve? Let$C$be some smooth proper curve of genus$g$over an algebraically closed field$k$. In order to understand special divisors on C one may consider the following function c(r), which I will call ... 1 vote 0 answers 144 views ### Replacing a non-basepoint free line bundle by a basepoint free one In his paper “Brill-noether-petri without degenerations”, Lazarsfeld says the following in in Corollary 1.4. Statement - Assume that every member of the linear series$|C_0|$in a K3 surface is ... • 643 4 votes 0 answers 166 views ### Brill-Noether statement for plane curves Let$C$be a smooth degree$d$curve in the plane. If$d$is large (at least 5 or so), then$C$is not Brill-Noether general because it admits a$g^2_d$. What, then, is the correct statement of Brill-... • 161 5 votes 0 answers 289 views ### Does the Torelli space appear "in nature"? What I mean by the (slightly facetious) title is: The classical theory of algebraic curves from the 19th century was split in two in the 20th century (much like the theory of groups): the theory of ... • 1,971 1 vote 0 answers 132 views ### Constructing embedded families of curves with general moduli Is there a known way to construct flat families of smooth curves$\mathcal{C}/\mathcal{B}$which are fiberwise embedded in a family of projective varieties$\mathcal{X}/\mathcal{B}$and which have ... • 1,971 2 votes 0 answers 142 views ### An analogue of Brill-Noether for hypersurfaces? Let$d,g,r$be natural numbers such that$d \geq 1$,$g \geq 2$,$r \geq 3$. Denote by$\mathcal{H}_{d,g,r}$the Hilbert scheme classifying subschemes of$\mathbb{P}^r$with Hilbert polynomial$P(x) = ...
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Background. Let $C$ be a smooth projective curve of genus $g$. Denote by $C^d$ its d-th symmetric product and by $G^r_d(C)$ its associated variety of linear series of type $\mathfrak{g}^r_d$, i.e.  ...