## Thesis errata

### Brian Osserman

This pages lists all current corrections to the submitted version of my PhD thesis, "Limit linear series in positive characteristic and Frobenius-unstable vector bundles on curves." Only corrections of mathematical consequence are listed, including typos which could cause confusion.

### Major corrections

The following corrections are the result of substantial errors in arguments which required some rethinking of arguments, and slight modifications of statements.
• Theorem I.5.1:
Assertions on degrees (and correspondingly, ramification at infinity) in statement slightly weakened to a reflect gap in the original argument; argument simplified as a result.
• Theorem II.4.3:
Original construction in the fifth paragraph for imposing vanishing along D in case (3) was incorrect, as the argument in the sixth paragraph for injectivity only worked over the original base, and not after base change. It is necessary to decompose D into D^Y and D^Z as in case (2), but this is not obviously possible while maintaining disjointness from the P_i. Hence, the argument for representability is made without imposing ramification, and the dimension count is performed etale locally, where we can work with D having the desired properties. The statement of Lemma II.4.2 is modified accordingly.
• Theorem III.0.1, Proposition III.6.4:
Not an error per se, but further calculations demonstrated that for the characteristic 7 explicit calculations, the equations h_{7,3} and h_{7,4} are redundant, so that in fact the space of connections with vanishing p-curvature can be cut out by 2 curves rather than 4. This then simplified certain other arguments as well.
• Section V.5:
beta_i must always be at most p - 2 alpha_i. However, the possibility that beta_i = p - 2 alpha_i had not been properly considered. For such i, it is necessary to specify the c_i in order to determine the connection. This affects the statements of Propositions V.5.5 and V.5.6 and Theorem V.5.7, but not Theorem V.0.1.

### Moderate corrections

The following corrections are the result of minor errors in arguments which required some modifications of statements and/or proofs.
• Proof of Corollary I.4.4:
Final paragraph added observing that the assertion on deformations of Theorem I.0.4 is also justified by the reduction argument.
• (III.1.2)-(III.1.3):
The formulas are only valid in the case that X is a curve, because otherwise not all connections are integrable. Also, it is necessary to assume that a connection exists on E. Hence, these formulas were moved after Theorem III.1.4.
• Proposition III.2.7:
Some confusion over permutation notation caused minor problems. This was corrected, as were minor typos in the statement and proof.
• Proof of Lemma IV.B.2, end:
m' could be a sum of products instead of a product, and is rewritten accordingly
• Proposition VI.2.7, Corollary VI.2.8:
In choosing a glued line bundle L, we must require that L^{otimes 2} = omega_C, as this is what will ensure that the glued E we produce will have trivial determinant.

### Minor corrections, statement-affecting

The following corrections are the result of typos affecting statements of results.
• Lemma II.4.2:
Statement is corrected to use the family of L^i rather than L, and modified to fit modified proof of Theorem II.4.3 (see major corrections).
• Remark II.4.8, end:
"yet we see that this does imply" -> "yet we see that this does not imply"
• Section III.7:
Replace psi by psi^0 throughout
• Situation IV.A.1:
"over a field" -> "over an algebraically closed field"
• Definition IV.B.4:
"every U" -> "every affine open U"
• Chapter V:
the epsilon in the statement of Theorem V.0.1, determining the parity of the degree of E, is replaced by delta here and throughout, to avoid conflict with epsilon used in deformations.
• Section V.4, first line:
"epsilon p \leq 2d" -> "delta p < 2d"
• Proposition V.4.5:
"In particular, in the case mp<d" -> "In particular, in the case (m+delta)p<d"
• Lemma A.13 (iii):
"in the sense that the generic point of M_1 is in the image" -> "in the sense that its image is not contained in any proper closed subscheme of M_1"

### Minor corrections, other

The following corrections are the result of typos not affecting statements of any results.
• Proof of Theorem I.2.3 (ii):
"a general fiber of the ramification morphism cannot have any pooints of MR' in its preimage" -> "a general fiber of the ramification morphism cannot contain any points of MR"
• Proof of Theorem I.A.5, final paragraph:
"is representable by a closed subscheme of Y" -> "is representable by a closed subscheme of X"
• Corollary II.4.4 (i):
"the generic fiber of Z over U" -> "the generic fiber of Z over B"
• Proof of Corollary II.4.4, end of second paragraph:
"some neighborhood U of b, as desired." -> "some neighborhood of b, as desired."
• Proof of Lemma II.5.3:
replace L^i_S and L^i_U by L^0_S and L^0_U throughout.
• Proof of Proposition II.5.7, middle of second paragraph:
"must be glued to a section of V^z_{d-i}" -> "must be glued to a section of V^z_i"
end of second paragraph:
"dim V^Z_{r-i} = r+2" -> "dim V^Z_i = r+2"
• Proof of Proposition II.5.8, middle of first paragraph:
"for some r, ... less than or equal to r ... less than r" -> "for some j, ... less than or equal to j ... less than j"
middle of third paragraph:
Replace L^Y by pi_* L^Y throughout.
Because the base scheme is arbitrary, remove "; in particular, it is torsion-free, and the cokernel of the first map must also be torsion-free", and note that the argument remains valid.
• Proof of Lemma II.A.10 (ii), middle of paragraph:
"if we choose \bar{v}_1, ... \bar{v}_{r_1} spanning \bar{g}_i(\bar{V}_{i+1}) and \bar{v}'_1, ... \bar{v}'_{r_2} spanning \bar{f}_i(\bar{V}_i)," -> "if we choose \bar{v}_1, ... \bar{v}_{r_1}\in \bar{V}_i such that the \bar{f}_i(\bar{v}_j) form a basis of \bar{f}_i(\bar{V}_i), and \bar{v}'_1, ... \bar{v}'_{r_2}\in \bar{V}_{i+1} such that the \bar{g}_i(\bar{v}'_j) form a basis of \bar{g}_i(\bar{V}_{i+1}),"
• Proof of Proposition III.2.5, end of first paragraph:
"\sum _{j<ell} \hat{n}_{\i_0-i_j}" -> "\sum _{j<ell} \hat{n}_{\i_0-1_j}"
"\sum j < ell" -> "\sum _{j < ell}"
• First paragraph after Situation III.3.1:
citation to [29] should be to Prop. 3.3, not 3.5
• First paragraph after proof of Proposition III.5.2:
"is invertible elsewhere on U_1, and omega_1 is invertible everywhere on U_1, (d phi_{12})/omega_1 is likewise everywhere invertible on U_1." -> "omega_1 is invertible everywhere on U_1, we likewise have (d phi_{12})/omega_1 invertible at w, and by further restriction of U_1 we may assume it is invertible on U_1."
• Remark III.5.8, end of third paragraph:
"subsheaf of F" -> "subsheaf of pi_* F"
"restricted to every fiber" -> "restricted to every point of the base"
• Proof of Lemma III.7.1, end of first paragraph:
Lower-right coefficient of matrix for T^p should be 0, not 1.
beginning of fourth paragraph:
"correspond to non-negative" -> "correspond to nonzero"
• Proof of Proposition IV.4.7, fourth and fifth paragraphs:
Replace H \otimes Omega^1_C by H^{-1} \otimes Omega^1_C throughout
• Proof of Lemma IV.5.3, fourth paragraph:
"in terms of which E, and E+E', are written" -> "in terms of which E and E' are written"
• Proof of Theorem IV.A.7, end of fourth paragraph:
"g n^2" -> "(g-1) n^2"
beginning of eighth paragraph:
"is itself finite flat" -> "is itself flat"
• Proof of Proposition IV.B.5, second paragraph:
"mapping to f y_l" -> "mapping to f n_1, ..., f n_l"
• Proof of Proposition V.1.1, beginning of second paragraph:
"quotient is torsion" -> "cokernel is torsion"
• Proof of Proposition V.2.3, beginning of third paragraph:
"s must itself be in E^nabla_P" -> "s must itself be in epsilon E^nabla_P"
• Section V.4, second paragraph:
Replace 2d and -2d in left-multiplication matrix by 2d - delta p and delta p - 2d respectively.
• Proof of Proposition V.4.1, end of first paragraph:
"and M_i by D(1, mu^{-1})" -> "and M_i by D(1, mu^{-1}) M_i"
middle of second paragraph:
"we can multiply M" -> "we can multiply M_i"
• Proof of Proposition V.5.6, middle of first paragraph:
Switch h_1, h_2 in expression for Delta. Switch \hat{g}_{11} and \hat{g}_{12} in following expressions for solutions to the same equation, and in displayed inequality in middle of following paragraph.
"changing q by a multiple of g_{21}" -> "changing q by a multiple of g_1"
• Proof of Proposition V.5.8, end of second paragraph:
"p - alpha_i" -> "p - 2 alpha_i"
• Section VI.4, first paragraph:
"of trivial determinant on E" -> "of trivial determinant on E_0"
• Proof of Proposition A.23:
Remove first sentence.
• Proof of Proposition A.30 (ii):
Take traces of all matrices in displayed equation.