We study the arithmetic properties of projective varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely
2
2
. We notably show, that such a variety
X
⊂
P
r
X \subset {\mathbb P}^r
is either arithmetically normal (and arithmetically Gorenstein) or a projection of a variety of minimal degree
X
~
⊂
P
r
+
1
\tilde {X} \subset {\mathbb P}^{r + 1}
from an appropriate point
p
∈
P
r
+
1
∖
X
~
p \in {\mathbb P}^{r + 1} \setminus \tilde {X}
. We focus on the latter situation and study
X
X
by means of the projection
X
~
→
X
\tilde {X} \rightarrow X
.
If
X
X
is not arithmetically Cohen-Macaulay, the homogeneous coordinate ring
B
B
of the projecting variety
X
~
\tilde {X}
is the endomorphism ring of the canonical module
K
(
A
)
K(A)
of the homogeneous coordinate ring
A
A
of
X
.
X.
If
X
X
is non-normal and is maximally Del Pezzo, that is, arithmetically Cohen-Macaulay but not arithmetically normal,
B
B
is just the graded integral closure of
A
.
A.
It turns out, that the geometry of the projection
X
~
→
X
\tilde {X} \rightarrow X
is governed by the arithmetic depth of
X
X
in any case.
We study, in particular, the case in which the projecting variety
X
~
⊂
P
r
+
1
\tilde {X} \subset {\mathbb P}^{r + 1}
is a (cone over a) rational normal scroll. In this case
X
X
is contained in a variety of minimal degree
Y
⊂
P
r
Y \subset {\mathbb P}^r
such that
codim
Y
(
X
)
=
1
\operatorname {codim}_Y(X) = 1
. We use this to approximate the Betti numbers of
X
X
.
In addition, we present several examples to illustrate our results and we draw some of the links to Fujita’s classification of polarized varieties of
Δ
\Delta
-genus
1
1
.