Mixing
Two questions about the proof of Clifford's theorem for compact Riemann surfaces.
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
I have two questions about the proof of Clifford's theorem in Algebraic Curves and Riemann Surfaces by Rick Miranda.
(a) In the beginning of the proof, a special divisor is defined as a divisor $D$ such that $D geq 0$ and $L(K - D) neq 0$. The author then writes
Note that $D$ is a special divisor if and only if both $dim L(D) geq 1$ and $dim L(K - D) geq 1$.
I don't see why the conditions $dim L(D) geq 1$ and $dim L(K - D) geq 1$ imply that $D geq 0$.
(b) A little later in the proof, we have a divisor $D$ and a linear system $| M |$ with no base points. The author notes that we can find a divisor $E in | M |$ such that $D$ and $E$ have disjoint support. If the support of $D$ consists of one point, this is of course trivially true, but I don't see why it holds if the support contains more points. Could anyone explain why this is so?
algebraic-curves riemann-surfaces
asked Jul 29 at 21:32
merle
536
add a comment |Â
up vote
2
down vote
favorite
I have two questions about the proof of Clifford's theorem in Algebraic Curves and Riemann Surfaces by Rick Miranda.
(a) In the beginning of the proof, a special divisor is defined as a divisor $D$ such that $D geq 0$ and $L(K - D) neq 0$. The author then writes
Note that $D$ is a special divisor if and only if both $dim L(D) geq 1$ and $dim L(K - D) geq 1$.
I don't see why the conditions $dim L(D) geq 1$ and $dim L(K - D) geq 1$ imply that $D geq 0$.
(b) A little later in the proof, we have a divisor $D$ and a linear system $| M |$ with no base points. The author notes that we can find a divisor $E in | M |$ such that $D$ and $E$ have disjoint support. If the support of $D$ consists of one point, this is of course trivially true, but I don't see why it holds if the support contains more points. Could anyone explain why this is so?
algebraic-curves riemann-surfaces
asked Jul 29 at 21:32
merle
536
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I have two questions about the proof of Clifford's theorem in Algebraic Curves and Riemann Surfaces by Rick Miranda.
(a) In the beginning of the proof, a special divisor is defined as a divisor $D$ such that $D geq 0$ and $L(K - D) neq 0$. The author then writes
Note that $D$ is a special divisor if and only if both $dim L(D) geq 1$ and $dim L(K - D) geq 1$.
I don't see why the conditions $dim L(D) geq 1$ and $dim L(K - D) geq 1$ imply that $D geq 0$.
(b) A little later in the proof, we have a divisor $D$ and a linear system $| M |$ with no base points. The author notes that we can find a divisor $E in | M |$ such that $D$ and $E$ have disjoint support. If the support of $D$ consists of one point, this is of course trivially true, but I don't see why it holds if the support contains more points. Could anyone explain why this is so?
algebraic-curves riemann-surfaces
asked Jul 29 at 21:32
merle
536
I have two questions about the proof of Clifford's theorem in Algebraic Curves and Riemann Surfaces by Rick Miranda.
(a) In the beginning of the proof, a special divisor is defined as a divisor $D$ such that $D geq 0$ and $L(K - D) neq 0$. The author then writes
Note that $D$ is a special divisor if and only if both $dim L(D) geq 1$ and $dim L(K - D) geq 1$.
I don't see why the conditions $dim L(D) geq 1$ and $dim L(K - D) geq 1$ imply that $D geq 0$.
(b) A little later in the proof, we have a divisor $D$ and a linear system $| M |$ with no base points. The author notes that we can find a divisor $E in | M |$ such that $D$ and $E$ have disjoint support. If the support of $D$ consists of one point, this is of course trivially true, but I don't see why it holds if the support contains more points. Could anyone explain why this is so?
algebraic-curves riemann-surfaces
asked Jul 29 at 21:32
merle
536
edited Jul 30 at 17:42
asked Jul 29 at 21:32
merle
536
asked Jul 29 at 21:32
merle
536
asked Jul 29 at 21:32
merle
536
536
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
I've solved both questions. For those who are interested:
(a) If $dim L(D) geq 1$, then $|D|$ is not empty, so there is a non-negative divisor which is linearly equivalent to $D$. So we can just assume $D geq 0$.
(b) Let $p_1, ldots, p_n$ be the support of $D$. Since $|M|$ has no base points, we can find a $g_i in L(M)$ such that $textord_p_i(g_i) + M(p_i) = 0$ for all $i$. Fix a local coordinate $z_i$ at every $p_i$. Now write $c_i$ for the $-M(p_i)$ coefficient of the Laurent expansion of $g_i$ in the coordinate $z_i$. Note that $c_ineq 0$. Then $h = sum_i frac1c_i g_i$ has the property that $textord_p_i(h) = textord_p_i(g_i)$ for all $i$, so $textdiv(h) + M$ has none of the $p_i$ in its support.
answered Jul 30 at 17:45
merle
536
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
I've solved both questions. For those who are interested:
(a) If $dim L(D) geq 1$, then $|D|$ is not empty, so there is a non-negative divisor which is linearly equivalent to $D$. So we can just assume $D geq 0$.
(b) Let $p_1, ldots, p_n$ be the support of $D$. Since $|M|$ has no base points, we can find a $g_i in L(M)$ such that $textord_p_i(g_i) + M(p_i) = 0$ for all $i$. Fix a local coordinate $z_i$ at every $p_i$. Now write $c_i$ for the $-M(p_i)$ coefficient of the Laurent expansion of $g_i$ in the coordinate $z_i$. Note that $c_ineq 0$. Then $h = sum_i frac1c_i g_i$ has the property that $textord_p_i(h) = textord_p_i(g_i)$ for all $i$, so $textdiv(h) + M$ has none of the $p_i$ in its support.
answered Jul 30 at 17:45
merle
536
add a comment |Â
up vote
0
down vote
accepted
I've solved both questions. For those who are interested:
(a) If $dim L(D) geq 1$, then $|D|$ is not empty, so there is a non-negative divisor which is linearly equivalent to $D$. So we can just assume $D geq 0$.
(b) Let $p_1, ldots, p_n$ be the support of $D$. Since $|M|$ has no base points, we can find a $g_i in L(M)$ such that $textord_p_i(g_i) + M(p_i) = 0$ for all $i$. Fix a local coordinate $z_i$ at every $p_i$. Now write $c_i$ for the $-M(p_i)$ coefficient of the Laurent expansion of $g_i$ in the coordinate $z_i$. Note that $c_ineq 0$. Then $h = sum_i frac1c_i g_i$ has the property that $textord_p_i(h) = textord_p_i(g_i)$ for all $i$, so $textdiv(h) + M$ has none of the $p_i$ in its support.
answered Jul 30 at 17:45
merle
536
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
I've solved both questions. For those who are interested:
(a) If $dim L(D) geq 1$, then $|D|$ is not empty, so there is a non-negative divisor which is linearly equivalent to $D$. So we can just assume $D geq 0$.
(b) Let $p_1, ldots, p_n$ be the support of $D$. Since $|M|$ has no base points, we can find a $g_i in L(M)$ such that $textord_p_i(g_i) + M(p_i) = 0$ for all $i$. Fix a local coordinate $z_i$ at every $p_i$. Now write $c_i$ for the $-M(p_i)$ coefficient of the Laurent expansion of $g_i$ in the coordinate $z_i$. Note that $c_ineq 0$. Then $h = sum_i frac1c_i g_i$ has the property that $textord_p_i(h) = textord_p_i(g_i)$ for all $i$, so $textdiv(h) + M$ has none of the $p_i$ in its support.
answered Jul 30 at 17:45
merle
536
I've solved both questions. For those who are interested:
(a) If $dim L(D) geq 1$, then $|D|$ is not empty, so there is a non-negative divisor which is linearly equivalent to $D$. So we can just assume $D geq 0$.
(b) Let $p_1, ldots, p_n$ be the support of $D$. Since $|M|$ has no base points, we can find a $g_i in L(M)$ such that $textord_p_i(g_i) + M(p_i) = 0$ for all $i$. Fix a local coordinate $z_i$ at every $p_i$. Now write $c_i$ for the $-M(p_i)$ coefficient of the Laurent expansion of $g_i$ in the coordinate $z_i$. Note that $c_ineq 0$. Then $h = sum_i frac1c_i g_i$ has the property that $textord_p_i(h) = textord_p_i(g_i)$ for all $i$, so $textdiv(h) + M$ has none of the $p_i$ in its support.
answered Jul 30 at 17:45
merle
536
answered Jul 30 at 17:45
merle
536
answered Jul 30 at 17:45
merle
536
answered Jul 30 at 17:45
merle
536
536
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2866458%2ftwo-questions-about-the-proof-of-cliffords-theorem-for-compact-riemann-surfaces%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password