Basic equine genetics for nonscientists...
First, some basic concepts:
1) Mendelian theory defined the 'dominant' and 'recessive' labels for genes based on inheritance patterns long before modern molecular biology existed.
2) Every horse has 2 copies of every gene.
3) Every foal inherits one copy of each gene from their sire and the complementary copy from their dam.
4) A gene is dominant if one copy from either parent is enough to cause a specific trait to be expressed, for example brown eyes.
5) A gene is recessive if two identical copies, one from each parent, are needed to cause the expression of a specific trait, for example blue eyes.
The existence of a 'dominant curly gene' has been well established by a dominant inheritance pattern in hundreds of pedigrees. In other words, you can breed a curly-coated horse carrying only one copy of the curly gene to a straight-coated horse and have a 50% chance of getting a curly foal...Mendelian genetics. If one parent has 2 copies of the dominant curly gene and the other is straight-coated, then all offspring will be curly, having inherited one copy of the dominant curly gene and one copy of the corresponding straight-coated gene.
If one parent carries the recessive curly gene (1 or 2 copies) and the other carries the dominant curly gene (1 or 2 copies), then the offspring could carry both mutations at the same gene locus (location). Likewise, traits encoded by the the recessive curly gene will only manifest themselves if the foal carries 2 copies of the recessive gene, one from each parent.
Part of the confusion in this field comes from the fact that neither gene has been DNA sequenced. It is very possible the dominant curly gene is a mutation in one gene that affects coat traits and that the recessive gene is a mutation in an entirely different gene that also affects coat traits. Then, just to make life more interesting, the expression of every gene (the traits you can see in an offspring, such as brown eyes) can be modified (changed) by other genes in the DNA make-up of each individual. This can lead to false assumptions that the underlying Mendelian inheritance isn't happening. There are hundreds of genes that affect the look and feel of a horse's coat, not just one or two.