The Gieseker–Petri theorem and imposed ramification

Chan, Melody; Osserman, Brian; Pflueger, Nathan

doi:10.1112/blms.12273

Cited by 8 publications

(22 citation statements)

References 17 publications

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“…Indeed, from the definition of c(i) (see §1.3), the entries of the two determinants in (13) do not match, but the determinants do. Furthermore, fix i with 1 ≤ i ≤ k t .…”

Section: General Casementioning

confidence: 99%

Motivic classes of degeneracy loci and pointed Brill-Noether varieties

Anderson¹,

Chen²,

Tarasca³

2020

Preprint

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Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch χy-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature. MSC2020. 14N15, 14H51 (primary), 19E99 (secondary).

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“…Indeed, from the definition of c(i) (see §1.3), the entries of the two determinants in (13) do not match, but the determinants do. Furthermore, fix i with 1 ≤ i ≤ k t .…”

Section: General Casementioning

confidence: 99%

Motivic classes of degeneracy loci and pointed Brill-Noether varieties

Anderson¹,

Chen²,

Tarasca³

2020

Preprint

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“…We are interested in generalizing Brion's result to a relative setting, motivated by Brill-Noether theory with imposed ramifications studied in [Oss11,CP17,COP19]. We ask the following question.…”

Section: Introductionmentioning

confidence: 99%

“…In the case when the base scheme is a 1-dimensional scheme, versality implies that the flags are transverse in most fibers but become almost transverse (i.e. exactly one pair of complementary dimensions of the two flags has a 1-dimensional intersection and the remaining pairs intersect trivially; see [COP19]) over finitely many reduced points. Their result generalizes [KWY13].…”

Section: Introductionmentioning

confidence: 99%

“…) defined with respect to versal flags; see [COP19,CP19]. Let λ = (λ i ) and λ = (λ i ) be partitions where…”

Section: Introductionmentioning

confidence: 99%

“…Let g ≥ 0, d > r ≥ 0, and let (X; P, Q) be twice-marked smooth projective curve. Denote by a • and b• sequences 0 ≤ a 1 < • • • < a r < a r+1 ≤ d and 0 ≤ b 1 < • • • < b r < b r+1 ≤ d.The moduli space of linear series of projective rank r of degree d with imposed ramifica-tion at P, Q prescribed by a • , b • respectively is the subscheme of the classical Brill-Noether variety G r d (X) G r d (X, (P, a • ), (Q, b • )) ⊆ G r d (X); see[COP19]. Chan, Osserman and Pflueger showed that in the case when g = 1 and P − Q is not a torsion point of order less than or equal to d in Pic 0 (X), the variety Gr d (X, (P, a • ), (Q, b • )) is the intersection of two relative Grassmannian Schubert varieties G r d (X, (P, a • )) and G r d (X, (Q, b • )) over the base scheme Pic d (X) (also see [CP19, Corollary…”

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confidence: 99%

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Relative Bott-Samelson varieties

2020

Preprint

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We prove that, defined with respect to versal flags, the product of two relative Bott-Samelson varieties over the flag bundle is a resolution of singularities of a relative Richardson variety. This result generalizes Brion's resolution of singularities of Richardson varieties to the relative setting [Bri05]. It reflects the phenomenon that the local geometry of a relative Richardson variety is completely governed by the two intersecting relative Schubert varieties, studied in [CP19]. We also prove an analogous theorem in the case of relative Grassmannian Richardson varieties, thereby furnishing a resolution of singularities for the Brill-Noether variety with imposed ramification on twice-marked elliptic curves.

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Brill-Noether loci with ramification at two points

Teixidor-i-Bigas

2022

Annali di Matematica

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Brill-Noether loci M r g,d are those subsets of the moduli space Mg determined by the existence of a linear series of degree d and dimension r. We provide a new proof of the non-emptiness of the loci when the expected codimension of the Brill-Noether loci satisfies 0 < −ρ(g, r, d) ≤ g − 2. We show that for a generic point of a component of this locus, the Petri map is onto and the maximal rank conjecture for quadrics is satisfied. We also show that Brill-Noether loci of the same codimension are distinct when the codimension is not too large Let C be a curve of genus g. The Brill-Noether locus G r d (C) parameterizes linear series of degree d and dimension r on C. When the Brill-Noether number ρ(g, r, d) ≥ 0, the Brill-Noether locus is non-empty on every curve and irreducible if the number is strictly positive. On the other hand, when the Brill-Noether number is negative, the locus is empty for C generic but may be non-empty on special curves. One can then define M r g,d as the locus of curves that posses a g r d .

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The Gieseker–Petri theorem and imposed ramification

Cited by 8 publications

References 17 publications

Motivic classes of degeneracy loci and pointed Brill-Noether varieties

Motivic classes of degeneracy loci and pointed Brill-Noether varieties

Relative Bott-Samelson varieties

Brill-Noether loci with ramification at two points

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