Citation: Clement Allan Tisdell. Renewable energy use and the renewable energy sector’s development: public finance, environmental externalities and sustainability[J]. Green Finance, 2019, 1(2): 156-173. doi: 10.3934/GF.2019.2.156
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Throughout the paper, we work over an algebraically closed field
Σk=Σk(C,L)⊆Pr |
of
Assume that
σk+1:Ck×C⟶Ck+1 |
be the morphism sending
Ek+1,L:=σk+1,∗p∗L, |
which is a locally free sheaf of rank
Bk(L):=P(Ek+1,L) |
equipped with the natural projection
H0(Bk(L),OBk(L)(1))=H0(Ck+1,Ek+1,)=H0(C,L), |
and therefore, the complete linear system
βk:Bk(L)⟶Pr=P(H0(C,L)). |
The
It is clear that there are natural inclusions
C=Σ0⊆Σ1⊆⋯⊆Σk−1⊆Σk⊆Pr. |
The preimage of
Theorem 1.1. Let
To prove the theorem, we utilize several line bundles defined on symmetric products of the curve. Let us recall the definitions here and refer the reader to [2] for further details. Let
Ck+1=C×⋯×C⏟k+1times |
be the
Ak+1,L:=Tk+1(L)(−2δk+1) |
be a line bundle on
The main ingredient in the proof of Theorem 1.1 is to study the positivity of the line bundle
Proposition 1.2. Let
In particular, if
In this section, we prove Theorem 1.1. We begin with showing Proposition 1.2.
Proof of Proposition 1.2. We proceed by induction on
Assume that
rz,k+1,L:H0(Ck+1,Ak+1,L)⟶H0(z,Ak+1,L|z) |
is surjective. We can choose a point
rz,k+1,L:H0(Ck+1,Ak+1,L)⟶H0(z,Ak+1,L|z) |
where all rows and columns are short exact sequences. By tensoring with
rz,k+1,L:H0(Ck+1,Ak+1,L)⟶H0(z,Ak+1,L|z) |
in which we use the fact that
Since
Lemma 2.1. Let
Proof. Note that
B′/A′⊗A′A′/m′q=B′/(m′qB′+A′)=B′/(m′p+A′)=0. |
By Nakayama lemma, we obtain
We keep using the notations used in the introduction. Recall that
αk,1:Bk−1(L)×C⟶Bk(L). |
To see it in details, we refer to [1,p.432,line –5]. We define the relative secant variety
Proposition 2.2. ([2,Proposition 3.15,Theorem 5.2,and Proposition 5.13]) Recall the situation described in the diagram
αk,1:Bk−1(L)×C⟶Bk(L). |
Let
1.
2.
3.
As a direct consequence of the above proposition, we have an identification
H0(Ck+1,Ak+1,L)=H0(Σk,IΣk−1|Σk(k+1)). |
We are now ready to give the proof of Theorem 1.1.
Proof of Theorem 1.1. Let
b:˜Σk:=BlΣk−1Σk⟶Σk |
be the blowup of
b:˜Σk:=BlΣk−1Σk⟶Σk |
We shall show that
Write
γ:˜Σk⟶P(V). |
On the other hand, one has an identification
ψ:Ck+1⟶P(V). |
Also note that
ψ:Ck+1⟶P(V). |
Take an arbitrary closed point
α−1(x)⊆π−1k(x″)∩β−1k(x′). |
However, the restriction of the morphism
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