# Questions tagged [brill-noether]

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### Do people study this generalized Brill-Noether theory for two sheaves

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 $...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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-...

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### 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 ...

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### 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 ...

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### 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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### De Jonquières formula vs. Relative GW invariants

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.
$$ ...