In this article we prove that for any complete toric variety, and for any Cartier divisor, the ring of global sections of multiples of the line bundle associated to the divisor is finitely generated.

Remark 1. We quickly recall a couple of definitions Let DivQ(X) := Div(X)⊗Q. On smooth projective surfaces all Q-Weil divisors are also Q-Cartier, hence we can write Q-Cartier every divisor D as a finite sum ∑ xiCi, where the Ci are distinct integral curves and xi ∈ Q. A divisor D is called effective (or sometimes positive) if xi ≥ 0 ∀i. If D · C ≥ 0 for all integral curves C then D we be calle...

In this note we review a simple criterion, due to Ekedahl, for superspecial curves defined over finite fields.Using this we generalize and give some simple proofs for some well-known superspecial curves.

We show the existence of (ϵ,n)-complements for (ϵ,R)-complementary projective generalized pairs (X,B+M) Fano type, when either coefficients B and μj belong to a finite set, or DCC set M′≡0, where M′=∑μjMj′ Mj′ are nef Cartier divisors.

We prove that the log Hodge de Rham spectral sequences of certain proper smooth schemes Cartier type in characteristic $$p>0$$ degenerate at $$E_1$$ . also Kodaira vanishings for them hold when they are projective. formulate weak Lefschetz conjecture crystalline cohomologies and it is true cases.

For a smooth scheme X of pure dimension d over field k and an effective Cartier divisor D⊂X whose support is simple normal crossing divisor, we construct cycle class map from the Chow group zero-cycles with modulus to cohomology relative Milnor K-sheaf.

For a smooth scheme X of pure dimension d over field k and an effective Cartier divisor D⊂X whose support is simple normal crossing divisor, we construct cycle class map from the Chow group zero-cycles with modulus to cohomology relative Milnor K-sheaf.